Pub Date : 2025-08-01Epub Date: 2025-08-27DOI: 10.1016/j.rinam.2025.100625
J. Noyola Rodriguez , Cynthia G. Esquer-Pérez , J.C. Hernández-Gómez , Omar Rosario Cayetano
We consider a generalization of KdV-type equations with a quartic nonlinearity (gKdV-4), which includes dissipation terms similar to those appearing in the Benjamin-Bona-Mahoney equation as well as in the well-known Camassa–Holm and Degasperis-Procesi equations. Our objective is to construct classical solitary wave solutions (solitons-antisolitons) to this equation.
{"title":"Smooth solitary waves for the generalized gKdV-4 equation","authors":"J. Noyola Rodriguez , Cynthia G. Esquer-Pérez , J.C. Hernández-Gómez , Omar Rosario Cayetano","doi":"10.1016/j.rinam.2025.100625","DOIUrl":"10.1016/j.rinam.2025.100625","url":null,"abstract":"<div><div>We consider a generalization of KdV-type equations with a quartic nonlinearity <span><math><msup><mrow><mi>u</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span> (gKdV-4), which includes dissipation terms similar to those appearing in the Benjamin-Bona-Mahoney equation as well as in the well-known Camassa–Holm and Degasperis-Procesi equations. Our objective is to construct classical solitary wave solutions (solitons-antisolitons) to this equation.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100625"},"PeriodicalIF":1.3,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144904587","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-01Epub Date: 2025-06-18DOI: 10.1016/j.rinam.2025.100602
Pengfei Luo , Yun Zhang , Lu Xu
The directional motivation of predator is influenced by the density of prey and its alarm call, this paper focuses on a three-species spatial intraguild predation model involving prey-taxis and alarm-taxis. By energy estimates and heat semigroup theory, we prove that this model possesses a bounded and global classical solution in -dimensional space () with Neumann boundary conditions.
{"title":"Global boundedness of a three-species spatial intraguild predation model with alarm-taxis","authors":"Pengfei Luo , Yun Zhang , Lu Xu","doi":"10.1016/j.rinam.2025.100602","DOIUrl":"10.1016/j.rinam.2025.100602","url":null,"abstract":"<div><div>The directional motivation of predator is influenced by the density of prey and its alarm call, this paper focuses on a three-species spatial intraguild predation model involving prey-taxis and alarm-taxis. By energy estimates and heat semigroup theory, we prove that this model possesses a bounded and global classical solution in <span><math><mi>N</mi></math></span>-dimensional space (<span><math><mrow><mi>N</mi><mo>≥</mo><mn>3</mn></mrow></math></span>) with Neumann boundary conditions.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100602"},"PeriodicalIF":1.4,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144307320","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-01Epub Date: 2025-08-20DOI: 10.1016/j.rinam.2025.100621
Louis Shuo Wang , Jiguang Yu
Most existing studies focus on environmental noise, with few studies focusing on pure demographic noise. We propose an analytical framework that applies stochastic differential equation tools to prove the well-posedness of solutions to such models with pure demographic noise and obtain moment and asymptotic bounds. We use this framework to prove that demographic noise does not lead to population extinction, and numerical results are consistent with it. Our proposed framework fills the gaps in research on the well-posedness and extinction impossibility of models with pure demographic noise and provides a rigorous mathematical framework for addressing a general ecology system in more sophisticated evolutionary setups.
{"title":"Analysis framework for stochastic predator–prey model with demographic noise","authors":"Louis Shuo Wang , Jiguang Yu","doi":"10.1016/j.rinam.2025.100621","DOIUrl":"10.1016/j.rinam.2025.100621","url":null,"abstract":"<div><div>Most existing studies focus on environmental noise, with few studies focusing on pure demographic noise. We propose an analytical framework that applies stochastic differential equation tools to prove the well-posedness of solutions to such models with pure demographic noise and obtain moment and asymptotic bounds. We use this framework to prove that demographic noise does not lead to population extinction, and numerical results are consistent with it. Our proposed framework fills the gaps in research on the well-posedness and extinction impossibility of models with pure demographic noise and provides a rigorous mathematical framework for addressing a general ecology system in more sophisticated evolutionary setups.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100621"},"PeriodicalIF":1.3,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144864565","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this study, we investigate the oscillatory behavior of a generalized Riemann–Weber type differential equation, incorporating a logarithmically varying perturbation term and a time delay. Specifically, we derive the precise conditions under which all non-trivial solutions of the considered equation oscillate when the effects of time delay and logarithmic perturbation act simultaneously. The oscillation constant, which determines the boundary between oscillatory and non-oscillatory behavior, coincides with that of the classical delayed equation. In particular, in the absence of a time delay, the generalized equation reduces to a known form, ensuring consistency with the existing theory.
{"title":"Oscillation of generalized Riemann–Weber type differential equations with delay","authors":"Kazuki Ishibashi , Shouki Miyauchi , Housei Sakikawa","doi":"10.1016/j.rinam.2025.100637","DOIUrl":"10.1016/j.rinam.2025.100637","url":null,"abstract":"<div><div>In this study, we investigate the oscillatory behavior of a generalized Riemann–Weber type differential equation, incorporating a logarithmically varying perturbation term and a time delay. Specifically, we derive the precise conditions under which all non-trivial solutions of the considered equation oscillate when the effects of time delay and logarithmic perturbation act simultaneously. The oscillation constant, which determines the boundary between oscillatory and non-oscillatory behavior, coincides with that of the classical delayed equation. In particular, in the absence of a time delay, the generalized equation reduces to a known form, ensuring consistency with the existing theory.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100637"},"PeriodicalIF":1.3,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145026361","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-01Epub Date: 2025-06-20DOI: 10.1016/j.rinam.2025.100605
Guoqiang Zhao, Dongxi Li
Cancer subtype analysis faces challenges due to limited availability of gene samples and the complexity of cancer gene expression data. The imbalance of Positive and negative category ratio and high-dimensional redundant information degrade prediction performance. This paper proposes an integrated extreme random forest with feature selection model TreeEM(Tree-enhanced Ensemble Model combining with feature selection) to enhance prediction ability and reduce computational costs. The TreeEM model combines the Max-Relevance and Min-Redundancy(MRMR) feature selection method with improved fusion undersampling random forest and extreme tree forest. The TreeEM model achieves excellent performance on three cancer datasets, especially on the multi-omics datasets BRCA(Breast Cancer) and ARCENE datasets, with average improvements of 7.90% and 1.90% in prediction accuracy, respectively. This model also uses TCGA data with known survival time for survival analysis and prediction, demonstrating the reliability of the TreeEM model. This work contributes to advancements in computational tools for cancer research, facilitating precision medicine approaches and improving decision-making. The above results provide new ideas for cancer subtype classification, but the existing methods still have limitations in data imbalance and high-dimensional feature processing. In the following section, the shortcomings of the current research and the innovative solutions of this paper are systematically described.
由于基因样本的有限可用性和癌症基因表达数据的复杂性，癌症亚型分析面临挑战。正负类比失衡和高维冗余信息会降低预测性能。为了提高预测能力和降低计算成本，本文提出了一种带有特征选择模型TreeEM（Tree-enhanced Ensemble model and feature selection）的集成极端随机森林模型。该模型将最大相关和最小冗余（MRMR）特征选择方法与改进的融合欠采样随机森林和极端树森林相结合。TreeEM模型在三个癌症数据集上取得了优异的表现，特别是在多组学数据集BRCA（Breast cancer）和ARCENE数据集上，预测准确率平均分别提高了7.90%和1.90%。该模型还使用已知生存时间的TCGA数据进行生存分析和预测，证明了TreeEM模型的可靠性。这项工作有助于癌症研究的计算工具的进步，促进精准医学方法和改进决策。上述结果为癌症亚型分类提供了新的思路，但现有方法在数据不平衡、高维特征处理等方面仍存在局限性。在接下来的部分中，系统地描述了当前研究的不足和本文的创新解决方案。
{"title":"TreeEM: Tree-enhanced ensemble model combining with feature selection for cancer subtype classification and survival prediction","authors":"Guoqiang Zhao, Dongxi Li","doi":"10.1016/j.rinam.2025.100605","DOIUrl":"10.1016/j.rinam.2025.100605","url":null,"abstract":"<div><div>Cancer subtype analysis faces challenges due to limited availability of gene samples and the complexity of cancer gene expression data. The imbalance of Positive and negative category ratio and high-dimensional redundant information degrade prediction performance. This paper proposes an integrated extreme random forest with feature selection model TreeEM(Tree-enhanced Ensemble Model combining with feature selection) to enhance prediction ability and reduce computational costs. The TreeEM model combines the Max-Relevance and Min-Redundancy(MRMR) feature selection method with improved fusion undersampling random forest and extreme tree forest. The TreeEM model achieves excellent performance on three cancer datasets, especially on the multi-omics datasets BRCA(Breast Cancer) and ARCENE datasets, with average improvements of 7.90% and 1.90% in prediction accuracy, respectively. This model also uses TCGA data with known survival time for survival analysis and prediction, demonstrating the reliability of the TreeEM model. This work contributes to advancements in computational tools for cancer research, facilitating precision medicine approaches and improving decision-making. The above results provide new ideas for cancer subtype classification, but the existing methods still have limitations in data imbalance and high-dimensional feature processing. In the following section, the shortcomings of the current research and the innovative solutions of this paper are systematically described.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100605"},"PeriodicalIF":1.4,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321581","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-01Epub Date: 2025-06-25DOI: 10.1016/j.rinam.2025.100606
HanLin Li, Jiang Zhou
This paper proves the weak type estimates of the Hardy–Littlewood maximal operator on local Morrey spaces associated with ball quasi-Banach function spaces. As an application, we further obtain the weak type estimates of the Hardy–Littlewood maximal operator on the local Morrey spaces with variable exponents and the homogeneous Herz-type spaces with variable exponents.
{"title":"Weak type estimates of Hardy–Littlewood maximal operator on local Morrey spaces associated with ball quasi-Banach function spaces","authors":"HanLin Li, Jiang Zhou","doi":"10.1016/j.rinam.2025.100606","DOIUrl":"10.1016/j.rinam.2025.100606","url":null,"abstract":"<div><div>This paper proves the weak type estimates of the Hardy–Littlewood maximal operator on local Morrey spaces associated with ball quasi-Banach function spaces. As an application, we further obtain the weak type estimates of the Hardy–Littlewood maximal operator on the local Morrey spaces with variable exponents and the homogeneous Herz-type spaces with variable exponents.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100606"},"PeriodicalIF":1.4,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144471263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-01Epub Date: 2025-07-09DOI: 10.1016/j.rinam.2025.100612
S. Zangoei Zadeh , M. Azizian , M. Sarvari
The Black–Scholes model, a powerful tool for valuation of equity options specially European equity options, is based on assumptions that are violated in some situations due to market realities. One of these cases is the instability of risk-free interest rates and the volatility of stock prices in the Black–Scholes model.
In this paper, in order to make the Black–Scholes model more in line with market realities, fixed parameters in the model, such as risk-free interest rates and stock price volatility, are considered with uncertainty. The obtained interval model is solved using discretization method and converting it into a minimization problem. Finally, The accuracy and efficiency of the method is tested by some numerical examples.
{"title":"An interval version of Black–Scholes European option pricing model and its numerical solution","authors":"S. Zangoei Zadeh , M. Azizian , M. Sarvari","doi":"10.1016/j.rinam.2025.100612","DOIUrl":"10.1016/j.rinam.2025.100612","url":null,"abstract":"<div><div>The Black–Scholes model, a powerful tool for valuation of equity options specially European equity options, is based on assumptions that are violated in some situations due to market realities. One of these cases is the instability of risk-free interest rates and the volatility of stock prices in the Black–Scholes model.</div><div>In this paper, in order to make the Black–Scholes model more in line with market realities, fixed parameters in the model, such as risk-free interest rates and stock price volatility, are considered with uncertainty. The obtained interval model is solved using discretization method and converting it into a minimization problem. Finally, The accuracy and efficiency of the method is tested by some numerical examples.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100612"},"PeriodicalIF":1.4,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144580081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-01Epub Date: 2025-08-25DOI: 10.1016/j.rinam.2025.100631
Kolade M. Owolabi , Edson Pindza , Eben Maré
This paper presents a unified and robust numerical framework that combines the Discrete Singular Convolution (DSC) method for spatial discretization with the Exponential Time Differencing Runge–Kutta (ETDRK4) scheme for temporal integration to solve reaction–diffusion systems. Specifically, we investigate the formation of Turing patterns – such as spots, stripes, and mixed structures – in classical models including the Gray–Scott, Brusselator, and Barrio–Varea–Aragón–Maini (BVAM) systems. The DSC method, employing the regularized Shannon’s delta kernel, delivers spectral-like accuracy in computing spatial derivatives on both regular and curved geometries. Coupled with the fourth-order ETDRK method, this approach enables efficient and stable time integration over long simulations. Importantly, we rigorously establish the necessary theoretical results – including convergence, stability, and consistency theorems, along with their proofs – for the combined DSC-ETDRK4 method when applied to both planar and curved surfaces. We demonstrate the capability of the proposed method to accurately reproduce and analyze complex spatiotemporal patterns on a variety of surfaces, including the plane, sphere, torus, and bumpy geometries. Numerical experiments confirm the method’s versatility, high accuracy, and computational efficiency, making it a powerful tool for the study of pattern formation in reaction–diffusion systems on diverse geometries.
{"title":"Turing patterns across geometries: A proven DSC-ETDRK4 solver from plane to sphere","authors":"Kolade M. Owolabi , Edson Pindza , Eben Maré","doi":"10.1016/j.rinam.2025.100631","DOIUrl":"10.1016/j.rinam.2025.100631","url":null,"abstract":"<div><div>This paper presents a unified and robust numerical framework that combines the Discrete Singular Convolution (DSC) method for spatial discretization with the Exponential Time Differencing Runge–Kutta (ETDRK4) scheme for temporal integration to solve reaction–diffusion systems. Specifically, we investigate the formation of Turing patterns – such as spots, stripes, and mixed structures – in classical models including the Gray–Scott, Brusselator, and Barrio–Varea–Aragón–Maini (BVAM) systems. The DSC method, employing the regularized Shannon’s delta kernel, delivers spectral-like accuracy in computing spatial derivatives on both regular and curved geometries. Coupled with the fourth-order ETDRK method, this approach enables efficient and stable time integration over long simulations. Importantly, we rigorously establish the necessary theoretical results – including convergence, stability, and consistency theorems, along with their proofs – for the combined DSC-ETDRK4 method when applied to both planar and curved surfaces. We demonstrate the capability of the proposed method to accurately reproduce and analyze complex spatiotemporal patterns on a variety of surfaces, including the plane, sphere, torus, and bumpy geometries. Numerical experiments confirm the method’s versatility, high accuracy, and computational efficiency, making it a powerful tool for the study of pattern formation in reaction–diffusion systems on diverse geometries.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100631"},"PeriodicalIF":1.3,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144893178","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-01Epub Date: 2025-07-01DOI: 10.1016/j.rinam.2025.100609
Soumia EL OMARI, Said MELLIANI
This paper addresses proving that solutions exist for nonlinear elliptic problems characterized by boundary conditions of non-local type, as well as their uniqueness, within the framework of weighted Sobolev spaces. These problems are motivated by applications in petroleum engineering, where non-local boundary conditions model complex interactions in stratified reservoirs with three-dimensional geometries. Using the properties of Leray–Lions type operators, compactness arguments, and a priori estimates, we establish a fundamental theorem guaranteeing the existence of weak solutions under suitable assumptions. A rigorous proof of the uniqueness of solutions is also provided by exploiting the strict monotonicity of the operator. This work expands the modeling capabilities for contexts where non-local interactions play a key role, offering relevant mathematical tools for simulating oil well performance and other similar applications.
{"title":"Boundary conditions of nonlocal type in weighted Sobolev spaces for nonlinear elliptic problems","authors":"Soumia EL OMARI, Said MELLIANI","doi":"10.1016/j.rinam.2025.100609","DOIUrl":"10.1016/j.rinam.2025.100609","url":null,"abstract":"<div><div>This paper addresses proving that solutions exist for nonlinear elliptic problems characterized by boundary conditions of non-local type, as well as their uniqueness, within the framework of weighted Sobolev spaces. These problems are motivated by applications in petroleum engineering, where non-local boundary conditions model complex interactions in stratified reservoirs with three-dimensional geometries. Using the properties of Leray–Lions type operators, compactness arguments, and a priori estimates, we establish a fundamental theorem guaranteeing the existence of weak solutions under suitable assumptions. A rigorous proof of the uniqueness of solutions is also provided by exploiting the strict monotonicity of the operator. This work expands the modeling capabilities for contexts where non-local interactions play a key role, offering relevant mathematical tools for simulating oil well performance and other similar applications.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100609"},"PeriodicalIF":1.4,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144517410","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-01Epub Date: 2025-08-27DOI: 10.1016/j.rinam.2025.100630
M.C. Rodríguez-Palánquex
This paper explores the properties of a family of absolutely irreducible projective plane curves, denoted , which are defined over a finite field of characteristic 2. The curves are explicitly given by the homogeneous equation , where and are natural numbers satisfying the conditions and . A primary objective of the paper is to determine the number of rational points on these curves.
The work also includes a detailed analysis of the singular points of the curves, providing a classification of these points based on the parameters and . Furthermore, the relationship between the number of rational points and the genus of the curves is investigated, with specific computations carried out for curves defined over the finite field . In particular, the paper presents explicit calculations of the number of rational points for curves of the form and over , illustrating the connection between these counts and the genus of the curves.
This comprehensive analysis contributes to a deeper understanding of the arithmetic geometry of this family of curves over finite fields.
{"title":"Rational and singular points of a family of curves","authors":"M.C. Rodríguez-Palánquex","doi":"10.1016/j.rinam.2025.100630","DOIUrl":"10.1016/j.rinam.2025.100630","url":null,"abstract":"<div><div>This paper explores the properties of a family of absolutely irreducible projective plane curves, denoted <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></msub></math></span>, which are defined over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> of characteristic 2. The curves are explicitly given by the homogeneous equation <span><math><mrow><msup><mrow><mi>Y</mi></mrow><mrow><mi>a</mi></mrow></msup><msup><mrow><mi>Z</mi></mrow><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></msup><mo>+</mo><mi>Y</mi><msup><mrow><mi>Z</mi></mrow><mrow><mi>b</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>b</mi></mrow></msup><mo>=</mo><mn>0</mn></mrow></math></span>, where <span><math><mi>a</mi></math></span> and <span><math><mi>b</mi></math></span> are natural numbers satisfying the conditions <span><math><mrow><mi>a</mi><mo>≥</mo><mn>2</mn></mrow></math></span> and <span><math><mrow><mi>b</mi><mo>≥</mo><mi>a</mi></mrow></math></span>. A primary objective of the paper is to determine the number of rational points on these curves.</div><div>The work also includes a detailed analysis of the singular points of the curves, providing a classification of these points based on the parameters <span><math><mi>a</mi></math></span> and <span><math><mi>b</mi></math></span>. Furthermore, the relationship between the number of rational points and the genus of the curves is investigated, with specific computations carried out for curves defined over the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>4</mn></mrow></msup></mrow></msub></math></span>. In particular, the paper presents explicit calculations of the number of rational points for curves of the form <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>b</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn><mo>,</mo><mi>b</mi></mrow></msub></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>4</mn></mrow></msup></mrow></msub></math></span>, illustrating the connection between these counts and the genus of the curves.</div><div>This comprehensive analysis contributes to a deeper understanding of the arithmetic geometry of this family of curves over finite fields.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100630"},"PeriodicalIF":1.3,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144904588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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