It is an elementary fact in the scientific literature that the Lipschitz norm of the realization function of a feedforward fully connected rectified linear unit (ReLU) artificial neural network (ANN) can, up to a multiplicative constant, be bounded from above by sums of powers of the norm of the ANN parameter vector. Roughly speaking, in this work we reveal in the case of shallow ANNs that the converse inequality is also true. More formally, we prove that the norm of the equivalence class of ANN parameter vectors with the same realization function is, up to a multiplicative constant, bounded from above by the sum of powers of the Lipschitz norm of the ANN realization function (with the exponents and ). Moreover, we prove that this upper bound only holds when employing the Lipschitz norm but does neither hold for Hölder norms nor for Sobolev-Slobodeckij norms. Furthermore, we prove that this upper bound only holds for sums of powers of the Lipschitz norm with the exponents and but does not hold for the Lipschitz norm alone.
{"title":"On Bounds for Norms of Reparameterized ReLU Artificial Neural Network Parameters: Sums of Fractional Powers of the Lipschitz Norm Control the Network Parameter Vector","authors":"Arnulf Jentzen, Timo Kröger","doi":"10.1002/mma.11080","DOIUrl":"https://doi.org/10.1002/mma.11080","url":null,"abstract":"<p>It is an elementary fact in the scientific literature that the Lipschitz norm of the realization function of a feedforward fully connected rectified linear unit (ReLU) artificial neural network (ANN) can, up to a multiplicative constant, be bounded from above by sums of powers of the norm of the ANN parameter vector. Roughly speaking, in this work we reveal in the case of shallow ANNs that the converse inequality is also true. More formally, we prove that the norm of the equivalence class of ANN parameter vectors with the same realization function is, up to a multiplicative constant, bounded from above by the sum of powers of the Lipschitz norm of the ANN realization function (with the exponents \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>/</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$$ 1/2 $$</annotation>\u0000 </semantics></math> and \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$$ 1 $$</annotation>\u0000 </semantics></math>). Moreover, we prove that this upper bound only holds when employing the Lipschitz norm but does neither hold for Hölder norms nor for Sobolev-Slobodeckij norms. Furthermore, we prove that this upper bound only holds for sums of powers of the Lipschitz norm with the exponents \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>/</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$$ 1/2 $$</annotation>\u0000 </semantics></math> and \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$$ 1 $$</annotation>\u0000 </semantics></math> but does not hold for the Lipschitz norm alone.</p>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 4","pages":"2135-2160"},"PeriodicalIF":1.8,"publicationDate":"2025-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mma.11080","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146154927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper introduces the adaptive finite difference image fusion (AFDIF) method, a novel partial differential equation (PDE)-based framework for enhancing multimodal image fusion. By employing adaptive grid refinement and stability-optimized finite difference discretization derived from a variational energy functional, AFDIF dynamically adjusts to local gradients, addressing limitations in fixed-grid PDE methods and improving feature preservation with 15% higher peak signal-to-noise ratio (PSNR), structural similarity index (SSIM) > 0.86, visual information fidelity (VIF) > 0.9, and gradient-based fusion metric > 0.7. Unlike established works, AFDIF fills recent knowledge gaps such as real-time scalability and hybrid deep learning integration through computational complexity and comparisons with deep learning methods. Experiments on public datasets and a proprietary dataset of 10 local medical pairs demonstrate superiority over baselines, validated by fivefold cross-validation, statistical -tests ( ), sensitivity analysis, and graphical validations. Key contributions include enhanced stability, real-time potential in diagnostics and remote sensing, and future deep learning-PDE hybrids. Its significance is as follows: advances computational imaging by bridging traditional and modern techniques.
本文介绍了一种新的基于偏微分方程（PDE）的图像融合框架——自适应有限差分图像融合（AFDIF）方法。通过采用自适应网格细化和由变分能量泛函导出的稳定性优化有限差分离散化，AFDIF动态调整到局部梯度，解决了固定网格PDE方法的局限性，并提高了15的特征保存% higher peak signal-to-noise ratio (PSNR), structural similarity index (SSIM) > 0.86, visual information fidelity (VIF) > 0.9, and gradient-based fusion metric Q A B / F $$ {Q}&#x0005E;{AB/F} $$ > 0.7. Unlike established works, AFDIF fills recent knowledge gaps such as real-time scalability and hybrid deep learning integration through computational complexity O ( N log N ) $$ Oleft(Nlog Nright) $$ and comparisons with deep learning methods. Experiments on public datasets and a proprietary dataset of 10 local medical pairs demonstrate superiority over baselines, validated by fivefold cross-validation, statistical t $$ t $$ -tests ( p < 0 . 01 $$ p<0.01 $$ ), sensitivity analysis, and graphical validations. Key contributions include enhanced stability, real-time potential in diagnostics and remote sensing, and future deep learning-PDE hybrids. Its significance is as follows: advances computational imaging by bridging traditional and modern techniques.
{"title":"Adaptive Finite Difference Image Fusion (AFDIF): A Novel PDE-Based Approach","authors":"Gargi Trivedi","doi":"10.1002/mma.70297","DOIUrl":"https://doi.org/10.1002/mma.70297","url":null,"abstract":"<div>\u0000 \u0000 <p>This paper introduces the adaptive finite difference image fusion (AFDIF) method, a novel partial differential equation (PDE)-based framework for enhancing multimodal image fusion. By employing adaptive grid refinement and stability-optimized finite difference discretization derived from a variational energy functional, AFDIF dynamically adjusts to local gradients, addressing limitations in fixed-grid PDE methods and improving feature preservation with 15% higher peak signal-to-noise ratio (PSNR), structural similarity index (SSIM) > 0.86, visual information fidelity (VIF) > 0.9, and gradient-based fusion metric \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mrow>\u0000 <mi>Q</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>A</mi>\u0000 <mi>B</mi>\u0000 <mo>/</mo>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$$ {Q}&amp;#x0005E;{AB/F} $$</annotation>\u0000 </semantics></math> > 0.7. Unlike established works, AFDIF fills recent knowledge gaps such as real-time scalability and hybrid deep learning integration through computational complexity \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>O</mi>\u0000 <mo>(</mo>\u0000 <mi>N</mi>\u0000 <mi>log</mi>\u0000 <mi>N</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$$ Oleft(Nlog Nright) $$</annotation>\u0000 </semantics></math> and comparisons with deep learning methods. Experiments on public datasets and a proprietary dataset of 10 local medical pairs demonstrate superiority over baselines, validated by fivefold cross-validation, statistical \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation>$$ t $$</annotation>\u0000 </semantics></math>-tests (\u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo><</mo>\u0000 <mn>0</mn>\u0000 <mo>.</mo>\u0000 <mn>01</mn>\u0000 </mrow>\u0000 <annotation>$$ p&lt;0.01 $$</annotation>\u0000 </semantics></math>), sensitivity analysis, and graphical validations. Key contributions include enhanced stability, real-time potential in diagnostics and remote sensing, and future deep learning-PDE hybrids. Its significance is as follows: advances computational imaging by bridging traditional and modern techniques.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 4","pages":"3127-3145"},"PeriodicalIF":1.8,"publicationDate":"2025-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146162750","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Eugenia Cornejo, Jesús Medina, Francisco José Ocaña
Modeling knowledge systems by determining relationships among key variables have been and currently is a fundamental and nontrivial challenge in real-world scenarios. Many approaches have been developed to reach this goal, but many of them are heuristic and require of alternative procedures to provide robust and tractable rules. With this significant aim, attribute implications were introduced in the mathematical framework of Formal Concept Analysis. In this paper, we will introduce a novel procedure to obtain relationships among variables from any dataset in which a Galois connection has been defined. In particular, we will be focused on the general multiadjoint framework, which is one of the most general approached in which a Galois connection have been considered, although the obtained results and methodology can also be used in other well-known approaches, such as in the residuated or heterogeneous frameworks.
{"title":"Attribute Implication Bases From Galois Connection Structures","authors":"M. Eugenia Cornejo, Jesús Medina, Francisco José Ocaña","doi":"10.1002/mma.70279","DOIUrl":"https://doi.org/10.1002/mma.70279","url":null,"abstract":"<p>Modeling knowledge systems by determining relationships among key variables have been and currently is a fundamental and nontrivial challenge in real-world scenarios. Many approaches have been developed to reach this goal, but many of them are heuristic and require of alternative procedures to provide robust and tractable rules. With this significant aim, attribute implications were introduced in the mathematical framework of Formal Concept Analysis. In this paper, we will introduce a novel procedure to obtain relationships among variables from any dataset in which a Galois connection has been defined. In particular, we will be focused on the general multiadjoint framework, which is one of the most general approached in which a Galois connection have been considered, although the obtained results and methodology can also be used in other well-known approaches, such as in the residuated or heterogeneous frameworks.</p>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 4","pages":"2729-2753"},"PeriodicalIF":1.8,"publicationDate":"2025-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mma.70279","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146197005","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Examining the extent to which measurements of rotation matrices are close to each other is challenging due measurement noise. To overcome this, data is typically smoothed, and the Riemannian and Euclidean metrics are applied. However, if rotation matrices are not directly measured and are instead formed by eigenvectors of measured symmetric matrices, this can be problematic if the associated eigenvalues are close. In this work, we propose novel semi-metrics that can be used to approximate the Riemannian metric for small angles. Our new results do not require eigenvector information and are beneficial for measured datasets. There are also issues when comparing rotational data arising from computational simulations, and it is important that the impact of the approximations on the computed outputs is properly assessed to ensure that the approximations made and the finite precision arithmetic are not unduly polluting the results. In this work, we examine data arising from object characterisation in metal detection using the complex symmetric rank two magnetic polarizability tensor (MPT) description; we rigorously analyze the effects of our numerical approximations and apply our new approximate measures of distance to the commutator of the real and imaginary parts of the MPT to this application. Our new approximate measures of distance provide additional feature information, which is invariant of the object orientation, to aid with object identification using machine learning classifiers. We present the Bayesian classification examples to demonstrate the success of our approach.
{"title":"How Far Are Two Symmetric Matrices From Commuting? With an Application to Object Characterisation and Identification in Metal Detection","authors":"P. D. Ledger, W. R. B. Lionheart, J. Elgy","doi":"10.1002/mma.70203","DOIUrl":"https://doi.org/10.1002/mma.70203","url":null,"abstract":"<p>Examining the extent to which measurements of rotation matrices are close to each other is challenging due measurement noise. To overcome this, data is typically smoothed, and the Riemannian and Euclidean metrics are applied. However, if rotation matrices are not directly measured and are instead formed by eigenvectors of measured symmetric matrices, this can be problematic if the associated eigenvalues are close. In this work, we propose novel semi-metrics that can be used to approximate the Riemannian metric for small angles. Our new results do not require eigenvector information and are beneficial for measured datasets. There are also issues when comparing rotational data arising from computational simulations, and it is important that the impact of the approximations on the computed outputs is properly assessed to ensure that the approximations made and the finite precision arithmetic are not unduly polluting the results. In this work, we examine data arising from object characterisation in metal detection using the complex symmetric rank two magnetic polarizability tensor (MPT) description; we rigorously analyze the effects of our numerical approximations and apply our new approximate measures of distance to the commutator of the real and imaginary parts of the MPT to this application. Our new approximate measures of distance provide additional feature information, which is invariant of the object orientation, to aid with object identification using machine learning classifiers. We present the Bayesian classification examples to demonstrate the success of our approach.</p>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 3","pages":"1914-1942"},"PeriodicalIF":1.8,"publicationDate":"2025-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mma.70203","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145986990","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is devoted to studying inverse source problems for the linear Kelvin–Voigt equations with memory, which govern the flow of incompressible viscoelastic fluids. The memory term is represented by an integral with convolution and reflects the elastic properties of the fluid. The inverse problems consist of recovering the spatial distribution of external forces by measurements for the velocity and gradient of pressure at the final moment . These inverse problems are investigated in the following cases: with and without the memory term in the momentum equation, and under both sticking and sliding boundary conditions. In all cases, under suitable conditions on the data of the problem, the existence, uniqueness as well as the stability of strong solutions were established.
本文研究了不可压缩粘弹性流体流动的具有记忆的线性Kelvin-Voigt方程的逆源问题。记忆项用卷积积分表示，反映了流体的弹性特性。反问题包括通过测量速度u $$ mathbf{u} $$和压力∇p的梯度来恢复外力f (x) $$ mathbf{f}(x) $$的空间分布$$ nabla p $$在最后一刻T $$ T $$。在动量方程中有和没有记忆项的情况下，以及在粘滞和滑动边界条件下，研究了这些反问题。在所有情况下，在问题数据的适当条件下，建立了强解的存在性、唯一性和稳定性。
{"title":"Inverse Source Problems for Kelvin–Voigt System With Final Overdetermination Condition","authors":"J. Cheng, Kh. Khompysh, N. Nugymanova","doi":"10.1002/mma.70308","DOIUrl":"https://doi.org/10.1002/mma.70308","url":null,"abstract":"<div>\u0000 \u0000 <p>This paper is devoted to studying inverse source problems for the linear Kelvin–Voigt equations with memory, which govern the flow of incompressible viscoelastic fluids. The memory term is represented by an integral with convolution and reflects the elastic properties of the fluid. The inverse problems consist of recovering the spatial distribution of external forces \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 <mo>(</mo>\u0000 <mi>x</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$$ mathbf{f}(x) $$</annotation>\u0000 </semantics></math> by measurements for the velocity \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>u</mi>\u0000 </mrow>\u0000 <annotation>$$ mathbf{u} $$</annotation>\u0000 </semantics></math> and gradient of pressure \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>∇</mo>\u0000 <mi>p</mi>\u0000 </mrow>\u0000 <annotation>$$ nabla p $$</annotation>\u0000 </semantics></math> at the final moment \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation>$$ T $$</annotation>\u0000 </semantics></math>. These inverse problems are investigated in the following cases: with and without the memory term in the momentum equation, and under both sticking and sliding boundary conditions. In all cases, under suitable conditions on the data of the problem, the existence, uniqueness as well as the stability of strong solutions were established.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 4","pages":"3317-3333"},"PeriodicalIF":1.8,"publicationDate":"2025-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146154976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this study, we focus on the detection of a vaccine-preventable communicable disease using a stochastic SIR compartmental model, which includes an additional compartment for individuals protected by vaccination. Although the vaccine is administered as a preventive measure before the onset of the disease, its efficacy diminishes with time, which requires booster doses for susceptible individuals. Using a continuous-time Markov chain, we examine the time required for disease detection (i.e., full confirmation of the disease) and analyze the incidence rates before detection, both within the vaccinated group and throughout the population. To illustrate the usefulness of our theoretical derivations and algorithmic schemes, we will present a numerical study involving the above descriptors for outbreaks of foot-and-mouth disease (FMD) in cattle.
{"title":"Incidence and Time Until Disease Detection in a Stochastic Model With Vaccination and General Transmission","authors":"Mariajesus Lopez-Herrero, Verdiana Mustaro, Diana Taipe","doi":"10.1002/mma.70306","DOIUrl":"https://doi.org/10.1002/mma.70306","url":null,"abstract":"<div>\u0000 \u0000 <p>In this study, we focus on the detection of a vaccine-preventable communicable disease using a stochastic SIR compartmental model, which includes an additional compartment for individuals protected by vaccination. Although the vaccine is administered as a preventive measure before the onset of the disease, its efficacy diminishes with time, which requires booster doses for susceptible individuals. Using a continuous-time Markov chain, we examine the time required for disease detection (i.e., full confirmation of the disease) and analyze the incidence rates before detection, both within the vaccinated group and throughout the population. To illustrate the usefulness of our theoretical derivations and algorithmic schemes, we will present a numerical study involving the above descriptors for outbreaks of foot-and-mouth disease (FMD) in cattle.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 4","pages":"3282-3302"},"PeriodicalIF":1.8,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146193563","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<p>The article examines a boundary-value problem in a bounded domain <span></span><math> <semantics> <mrow> <msub> <mrow> <mi>Ω</mi> </mrow> <mrow> <mi>ε</mi> </mrow> </msub> </mrow> <annotation>$$ {Omega}_{varepsilon } $$</annotation> </semantics></math> consisting of perforated and imperforate regions, with Neumann conditions prescribed at the boundaries of the perforations. Assuming the porous medium has symmetric, periodic structure with a small period <span></span><math> <semantics> <mrow> <mi>ε</mi> </mrow> <annotation>$$ varepsilon $$</annotation> </semantics></math>, we analyze the limit behavior of the problem as <span></span><math> <semantics> <mrow> <mi>ε</mi> <mo>→</mo> <mn>0</mn> </mrow> <annotation>$$ varepsilon to 0 $$</annotation> </semantics></math>. A crucial aspect of this study is deriving correct coupling conditions at the common interface, which is achieved using inner-layer asymptotics. For the flat interface, we construct and justify a complete asymptotic expansion of the solution in the Sobolev space <span></span><math> <semantics> <mrow> <msup> <mrow> <mi>H</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msup> <mo>(</mo> <msub> <mrow> <mi>Ω</mi> </mrow> <mrow> <mi>ε</mi> </mrow> </msub> <mo>)</mo> </mrow> <annotation>$$ {H}&#x0005E;1left({Omega}_{varepsilon}right) $$</annotation> </semantics></math>. Furthermore, for the <span></span><math> <semantics> <mrow> <mi>ε</mi> </mrow> <annotation>$$ varepsilon $$</annotation> </semantics></math>-periodically oscillating interface of amplitude <span></span><math> <mrow> <mi>𝒪</mi> <mo>(</mo> <mi>ε</mi> <mo>)</mo> </mrow></math>, we provide an approximation to the solution, detect the impact of the interface in the second term of the asymptotics, and
{"title":"Inner-Layer Asymptotics in Partially Perforated Domains: Coupling Across Flat and Oscillating Interfaces","authors":"Taras Melnyk","doi":"10.1002/mma.70310","DOIUrl":"https://doi.org/10.1002/mma.70310","url":null,"abstract":"<p>The article examines a boundary-value problem in a bounded domain \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mrow>\u0000 <mi>Ω</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>ε</mi>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$$ {Omega}_{varepsilon } $$</annotation>\u0000 </semantics></math> consisting of perforated and imperforate regions, with Neumann conditions prescribed at the boundaries of the perforations. Assuming the porous medium has symmetric, periodic structure with a small period \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ε</mi>\u0000 </mrow>\u0000 <annotation>$$ varepsilon $$</annotation>\u0000 </semantics></math>, we analyze the limit behavior of the problem as \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ε</mi>\u0000 <mo>→</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$$ varepsilon to 0 $$</annotation>\u0000 </semantics></math>. A crucial aspect of this study is deriving correct coupling conditions at the common interface, which is achieved using inner-layer asymptotics. For the flat interface, we construct and justify a complete asymptotic expansion of the solution in the Sobolev space \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msup>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mrow>\u0000 <mi>Ω</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>ε</mi>\u0000 </mrow>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$$ {H}&amp;#x0005E;1left({Omega}_{varepsilon}right) $$</annotation>\u0000 </semantics></math>. Furthermore, for the \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ε</mi>\u0000 </mrow>\u0000 <annotation>$$ varepsilon $$</annotation>\u0000 </semantics></math>-periodically oscillating interface of amplitude \u0000<span></span><math>\u0000 <mrow>\u0000 <mi>𝒪</mi>\u0000 <mo>(</mo>\u0000 <mi>ε</mi>\u0000 <mo>)</mo>\u0000 </mrow></math>, we provide an approximation to the solution, detect the impact of the interface in the second term of the asymptotics, and ","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 4","pages":"3353-3384"},"PeriodicalIF":1.8,"publicationDate":"2025-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mma.70310","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146162785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Wei Shi, Jiaquan Xie, Xinlong Guo, Jialin Si, Mingxu Yi
In this paper, the dynamical characteristics of nonlinear fractional van der Pol–Duffing systems are studied. The harmonic balance method is utilized to determine the main resonance of the system, and the approximate analytical solution of 1/3 subharmonic resonance of the system is calculated by means of the averaging method. The analytical expressions of amplitude-frequency and phase-frequency characteristic curves based on the harmonic balance method and the averaging method are established. The analytical solutions of the system are analyzed by Routh–Hurwitz and the stability conditions of the system are obtained. The numerical simulation is used to compare the analytical solution with the numerical solution. The results show that the numerical solution and the analytical solution are consistent in trend. The paper focuses on the research and analysis of the fork bifurcation and saddle-node bifurcation of the system and explores the influence of cubic stiffness, quintic stiffness, linear damping, nonlinear damping, and external excitation amplitude on the bifurcation law of the system. Finally, we analyze the chaos threshold under external excitation, as well as the time history diagram, phase trajectory and Poincaré section of the system with and without chaos, so as to verify the correctness of the derivation of chaos theory in this paper. The results indicate that as the excitation frequency and amplitude change, the system enters chaotic motion through a dual period bifurcation.
本文研究了非线性分数阶van der Pol-Duffing系统的动力学特性。采用谐波平衡法确定了系统的主共振，并采用平均法计算了系统1/3次谐波共振的近似解析解。建立了基于谐波平衡法和平均法的幅频和相频特性曲线解析表达式。采用Routh-Hurwitz方法对系统的解析解进行了分析，得到了系统的稳定条件。通过数值模拟对解析解与数值解进行了比较。结果表明，数值解与解析解在趋势上是一致的。本文重点研究和分析了系统的叉分岔和鞍节分岔，探讨了三次刚度、五次刚度、线性阻尼、非线性阻尼和外部激励幅值对系统分岔规律的影响。最后，分析了外部激励下的混沌阈值，以及有混沌和无混沌系统的时程图、相轨迹和庞卡罗剖面，验证了本文混沌理论推导的正确性。结果表明，随着激励频率和幅值的变化，系统通过双周期分岔进入混沌运动。
{"title":"Subharmonic Resonance of Fractional-Quintic van der Pol–Duffing Oscillator","authors":"Wei Shi, Jiaquan Xie, Xinlong Guo, Jialin Si, Mingxu Yi","doi":"10.1002/mma.70309","DOIUrl":"https://doi.org/10.1002/mma.70309","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, the dynamical characteristics of nonlinear fractional van der Pol–Duffing systems are studied. The harmonic balance method is utilized to determine the main resonance of the system, and the approximate analytical solution of 1/3 subharmonic resonance of the system is calculated by means of the averaging method. The analytical expressions of amplitude-frequency and phase-frequency characteristic curves based on the harmonic balance method and the averaging method are established. The analytical solutions of the system are analyzed by Routh–Hurwitz and the stability conditions of the system are obtained. The numerical simulation is used to compare the analytical solution with the numerical solution. The results show that the numerical solution and the analytical solution are consistent in trend. The paper focuses on the research and analysis of the fork bifurcation and saddle-node bifurcation of the system and explores the influence of cubic stiffness, quintic stiffness, linear damping, nonlinear damping, and external excitation amplitude on the bifurcation law of the system. Finally, we analyze the chaos threshold under external excitation, as well as the time history diagram, phase trajectory and Poincaré section of the system with and without chaos, so as to verify the correctness of the derivation of chaos theory in this paper. The results indicate that as the excitation frequency and amplitude change, the system enters chaotic motion through a dual period bifurcation.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 4","pages":"3334-3352"},"PeriodicalIF":1.8,"publicationDate":"2025-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146176465","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Beltrán de la Flor, Adrián Ruiz, Concepción Muriel
Liénard equations are analyzed using the recent theory of -structures. For each Liénard equation, a -structure is determined by using a Lie point symmetry and a -symmetry. Based on this approach, a novel method for integrating these equations is proposed, which consists in solving sequentially two completely integrable Pfaffian equations. The first Pfaffian equation can be solved by quadrature. For a class of Liénard equations admitting a certain -symmetry, this primitive is explicitly determined. The integration procedure is then completed by solving a one-parameter family of first-order equations. Additionally, for some subclasses of Liénard equations, an extra one-parameter family of solutions can be obtained via quadrature. These results are used to derive new exact solutions for a nonlinear Klein–Gordon equation with variable coefficients.
{"title":"C∞-Structures for Liénard Equations and New Exact Solutions to a Class of Klein–Gordon Equations","authors":"Beltrán de la Flor, Adrián Ruiz, Concepción Muriel","doi":"10.1002/mma.70282","DOIUrl":"https://doi.org/10.1002/mma.70282","url":null,"abstract":"<p>Liénard equations are analyzed using the recent theory of \u0000<span></span><math>\u0000 <mrow>\u0000 <msup>\u0000 <mrow>\u0000 <mi>𝒞</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 </msup>\u0000 </mrow></math>-structures. For each Liénard equation, a \u0000<span></span><math>\u0000 <mrow>\u0000 <msup>\u0000 <mrow>\u0000 <mi>𝒞</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 </msup>\u0000 </mrow></math>-structure is determined by using a Lie point symmetry and a \u0000<span></span><math>\u0000 <mrow>\u0000 <msup>\u0000 <mrow>\u0000 <mi>𝒞</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 </msup>\u0000 </mrow></math>-symmetry. Based on this approach, a novel method for integrating these equations is proposed, which consists in solving sequentially two completely integrable Pfaffian equations. The first Pfaffian equation can be solved by quadrature. For a class of Liénard equations admitting a certain \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$$ {C}&amp;amp;#x0005E;{infty } $$</annotation>\u0000 </semantics></math>-symmetry, this primitive is explicitly determined. The integration procedure is then completed by solving a one-parameter family of first-order equations. Additionally, for some subclasses of Liénard equations, an extra one-parameter family of solutions can be obtained via quadrature. These results are used to derive new exact solutions for a nonlinear Klein–Gordon equation with variable coefficients.</p>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 4","pages":"2795-2822"},"PeriodicalIF":1.8,"publicationDate":"2025-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mma.70282","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146162786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Truong Minh Tuyen, Nguyen Song Ha, Simeon Reich, Pham Thi Minh Hanh
We investigate the split mixed variational inequality problem with multiple output sets in real Hilbert spaces. By incorporating multiple inertial steps, utilizing a general index control mapping, and employing either hybrid methods or shrinking projection techniques, we introduce two novel cyclic projection algorithms for solving this problem without requiring the inverse strong monotonicity assumption on the associated operators. In addition, the control parameters are designed to be self-adaptive.
{"title":"Two Novel Cyclic Projection Algorithms for Solving Split Mixed Variational Inequalities Beyond Co-Coerciveness","authors":"Truong Minh Tuyen, Nguyen Song Ha, Simeon Reich, Pham Thi Minh Hanh","doi":"10.1002/mma.70300","DOIUrl":"https://doi.org/10.1002/mma.70300","url":null,"abstract":"<div>\u0000 \u0000 <p>We investigate the split mixed variational inequality problem with multiple output sets in real Hilbert spaces. By incorporating multiple inertial steps, utilizing a general index control mapping, and employing either hybrid methods or shrinking projection techniques, we introduce two novel cyclic projection algorithms for solving this problem without requiring the inverse strong monotonicity assumption on the associated operators. In addition, the control parameters are designed to be self-adaptive.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 4","pages":"3186-3210"},"PeriodicalIF":1.8,"publicationDate":"2025-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146154837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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