This paper presents a comprehensive study of the linguistic Z-graph, which is a novel framework designed to analyze linguistic structures within social networks. By integrating concepts from graph theory and linguistics, the linguistic Z-graph provides a detailed understanding of language dynamics in online communities. This study highlights the practical applications of linguistic Z-graphs in identifying central nodes within social networks, which are crucial for online businesses in market capture and information dissemination. Traditional methods for identifying central nodes rely on direct connections, but social network connections often exhibit uncertainty. This paper focuses on using fuzzy theory, particularly linguistic Z-graphs, to address this uncertainty, offering more detailed insights compared to fuzzy graphs. Our study introduces a new centrality measure using linguistic Z-graphs, enhancing our understanding of social network structures.
本文介绍了对语言 Z 图的全面研究，语言 Z 图是一个新颖的框架，旨在分析社交网络中的语言结构。通过整合图论和语言学的概念，语言 Z 图提供了对网络社区中语言动态的详细了解。本研究强调了语言 Z 图在识别社交网络中心节点方面的实际应用，而中心节点对于在线企业的市场占领和信息传播至关重要。识别中心节点的传统方法依赖于直接连接，但社交网络连接往往表现出不确定性。本文主要利用模糊理论，特别是语言 Z 图来解决这种不确定性，与模糊图相比，它能提供更详细的见解。我们的研究利用语言 Z 图引入了一种新的中心性度量方法，增强了我们对社交网络结构的理解。
{"title":"A Study on Linguistic Z-Graph and Its Application in Social Networks","authors":"Rupkumar Mahapatra, Sovan Samanta, Madhumangal Pal, Tofigh Allahviranloo, Antonios Kalampakas","doi":"10.3390/math12182898","DOIUrl":"https://doi.org/10.3390/math12182898","url":null,"abstract":"This paper presents a comprehensive study of the linguistic Z-graph, which is a novel framework designed to analyze linguistic structures within social networks. By integrating concepts from graph theory and linguistics, the linguistic Z-graph provides a detailed understanding of language dynamics in online communities. This study highlights the practical applications of linguistic Z-graphs in identifying central nodes within social networks, which are crucial for online businesses in market capture and information dissemination. Traditional methods for identifying central nodes rely on direct connections, but social network connections often exhibit uncertainty. This paper focuses on using fuzzy theory, particularly linguistic Z-graphs, to address this uncertainty, offering more detailed insights compared to fuzzy graphs. Our study introduces a new centrality measure using linguistic Z-graphs, enhancing our understanding of social network structures.","PeriodicalId":18303,"journal":{"name":"Mathematics","volume":"144 3 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249109","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 the design of real-world networks, researchers evaluate various structural parameters to assess vulnerability, including connectivity, toughness, and tenacity. Recently, the tightness metric has emerged as a potentially superior vulnerability measure, although many related theorems remain unknown due to its novelty. Harary graphs, known for their maximum connectivity, are an important class of graph models for network design. Prior work has evaluated the vulnerability of three types of Harary graphs using different parameters, but the tightness metric has not been thoroughly explored. This article aims to calculate the tightness values for all three types of Harary graphs. First, it will attempt to calculate the lower bound for the value of the tightness parameter in Harary graphs using existing lemmas and theorems. Then, by presenting new lemmas and theorems, we will try to find the exact value or upper bound for this parameter in Harary graphs. For the first type of Harary graph, the tightness is precisely determined, while for the second and third types, upper bounds are provided due to structural complexity. The lemmas, theorems, and proof methods presented in this research may be used to calculate other graph and network parameters. However, the newness of the tightness parameter means that further research is needed to fully characterize its properties.
{"title":"Tightness of Harary Graphs","authors":"Abolfazl Javan, Ali Moeini, Mohammad Shekaramiz","doi":"10.3390/math12182894","DOIUrl":"https://doi.org/10.3390/math12182894","url":null,"abstract":"In the design of real-world networks, researchers evaluate various structural parameters to assess vulnerability, including connectivity, toughness, and tenacity. Recently, the tightness metric has emerged as a potentially superior vulnerability measure, although many related theorems remain unknown due to its novelty. Harary graphs, known for their maximum connectivity, are an important class of graph models for network design. Prior work has evaluated the vulnerability of three types of Harary graphs using different parameters, but the tightness metric has not been thoroughly explored. This article aims to calculate the tightness values for all three types of Harary graphs. First, it will attempt to calculate the lower bound for the value of the tightness parameter in Harary graphs using existing lemmas and theorems. Then, by presenting new lemmas and theorems, we will try to find the exact value or upper bound for this parameter in Harary graphs. For the first type of Harary graph, the tightness is precisely determined, while for the second and third types, upper bounds are provided due to structural complexity. The lemmas, theorems, and proof methods presented in this research may be used to calculate other graph and network parameters. However, the newness of the tightness parameter means that further research is needed to fully characterize its properties.","PeriodicalId":18303,"journal":{"name":"Mathematics","volume":"20 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249105","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 paper, we consider the Cauchy problem of a time-fractional nonlinear diffusion equation. According to Kaplan’s first eigenvalue method, we first prove the blow-up of the solutions in finite time under some sufficient conditions. We next provide sufficient conditions for the existence of global solutions by using the results of Zhang and Sun. In conclusion, we find the second critical exponent for the existence of global and non-global solutions via the decay rates of the initial data at spatial infinity.
本文考虑了时间分数非线性扩散方程的 Cauchy 问题。根据 Kaplan 的第一特征值方法，我们首先证明了在某些充分条件下有限时间内解的炸毁。接下来，我们利用 Zhang 和 Sun 的结果为全局解的存在提供了充分条件。最后，我们通过空间无穷大处初始数据的衰减率，找到了全局解和非全局解存在的第二个临界指数。
{"title":"The Second Critical Exponent for a Time-Fractional Reaction-Diffusion Equation","authors":"Takefumi Igarashi","doi":"10.3390/math12182895","DOIUrl":"https://doi.org/10.3390/math12182895","url":null,"abstract":"In this paper, we consider the Cauchy problem of a time-fractional nonlinear diffusion equation. According to Kaplan’s first eigenvalue method, we first prove the blow-up of the solutions in finite time under some sufficient conditions. We next provide sufficient conditions for the existence of global solutions by using the results of Zhang and Sun. In conclusion, we find the second critical exponent for the existence of global and non-global solutions via the decay rates of the initial data at spatial infinity.","PeriodicalId":18303,"journal":{"name":"Mathematics","volume":"14 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249106","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}
Caused by nonlinear vibration, ocean white noise exhibits complex dynamic characteristics and nonlinear perception characteristics. To explore the potential application of ocean white noise in engineering and health fields, novel methods based on deep learning algorithms are proposed to generate ocean white noise, contributing to marine environment simulation in ocean engineering. A comparative study, including spectrum analysis and auditory testing, proved the superiority of the generation method using deep learning networks over general mathematical or physical methods. To further study the nonlinear perception characteristics of ocean white noise, novel experimental research based on multi-modal perception research methods was carried out within a constructed multi-modal perception system environment, including the following two experiments. The first audiovisual comparative experiment thoroughly explores the system’s user multi-modal perception experience and influence factors, explicitly focusing on the impact of ocean white noise on human perception. The second sound intensity testing experiment is conducted to further explore human multi-sensory interaction and change patterns under white noise stimulation. The experimental results indicate that user visual perception ability and state reach a relatively high level when the sound intensity is close to 50 dB. Further numerical analysis based on the experimental results reveals the internal influence relationship between user perception of multiple senses, showing a fluctuating influence law to user visual concentration and a curvilinear influence law to user visual psychology from the sound intensity of ocean white noise. This study underscores ocean white noise’s positive effect on human perception enhancement and concentration improvement, providing a research basis for multiple field applications such as spiritual healing, perceptual learning, and artistic creation for human beings. Importantly, it provides valuable references and practical insights for professionals in related fields, contributing to the development and utilization of the marine environment.
海洋白噪声由非线性振动引起，具有复杂的动态特性和非线性感知特性。为了探索海洋白噪声在工程和健康领域的潜在应用，本文提出了基于深度学习算法生成海洋白噪声的新方法，为海洋工程中的海洋环境模拟做出了贡献。包括频谱分析和听觉测试在内的比较研究证明，利用深度学习网络生成的方法优于一般的数学或物理方法。为进一步研究海洋白噪声的非线性感知特征，在构建的多模态感知系统环境中，基于多模态感知研究方法开展了新颖的实验研究，包括以下两个实验。第一个视听对比实验深入探讨系统的用户多模态感知体验和影响因素，明确关注海洋白噪声对人类感知的影响。第二个声强测试实验是为了进一步探索白噪声刺激下的人类多感官交互和变化规律。实验结果表明，当声强接近 50 dB 时，用户的视觉感知能力和状态达到了相对较高的水平。基于实验结果的进一步数值分析揭示了用户多感官感知的内在影响关系，显示了海洋白噪声声强对用户视觉集中度的波动影响规律和对用户视觉心理的曲线影响规律。这项研究强调了海洋白噪声对人类感知增强和注意力提高的积极作用，为人类精神治疗、感知学习和艺术创作等多个领域的应用提供了研究基础。重要的是，它为相关领域的专业人士提供了有价值的参考和实用见解，为海洋环境的开发和利用做出了贡献。
{"title":"Nonlinear Perception Characteristics Analysis of Ocean White Noise Based on Deep Learning Algorithms","authors":"Tao Qian, Ying Li, Jun Chen","doi":"10.3390/math12182892","DOIUrl":"https://doi.org/10.3390/math12182892","url":null,"abstract":"Caused by nonlinear vibration, ocean white noise exhibits complex dynamic characteristics and nonlinear perception characteristics. To explore the potential application of ocean white noise in engineering and health fields, novel methods based on deep learning algorithms are proposed to generate ocean white noise, contributing to marine environment simulation in ocean engineering. A comparative study, including spectrum analysis and auditory testing, proved the superiority of the generation method using deep learning networks over general mathematical or physical methods. To further study the nonlinear perception characteristics of ocean white noise, novel experimental research based on multi-modal perception research methods was carried out within a constructed multi-modal perception system environment, including the following two experiments. The first audiovisual comparative experiment thoroughly explores the system’s user multi-modal perception experience and influence factors, explicitly focusing on the impact of ocean white noise on human perception. The second sound intensity testing experiment is conducted to further explore human multi-sensory interaction and change patterns under white noise stimulation. The experimental results indicate that user visual perception ability and state reach a relatively high level when the sound intensity is close to 50 dB. Further numerical analysis based on the experimental results reveals the internal influence relationship between user perception of multiple senses, showing a fluctuating influence law to user visual concentration and a curvilinear influence law to user visual psychology from the sound intensity of ocean white noise. This study underscores ocean white noise’s positive effect on human perception enhancement and concentration improvement, providing a research basis for multiple field applications such as spiritual healing, perceptual learning, and artistic creation for human beings. Importantly, it provides valuable references and practical insights for professionals in related fields, contributing to the development and utilization of the marine environment.","PeriodicalId":18303,"journal":{"name":"Mathematics","volume":"51 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249101","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}
Accurate forecasting of high-resolution particulate matter 2.5 (PM2.5) levels is essential for the development of public health policy. However, datasets used for this purpose often contain missing observations. This study presents a two-stage approach to handle this problem. The first stage is a multivariate spatial time series (MSTS) model, used to generate forecasts for the sampled spatial units and to impute missing observations. The MSTS model utilizes the similarities between the temporal patterns of the time series of the spatial units to impute the missing data across space. The second stage is the high-resolution prediction model, which generates predictions that cover the entire study domain. The second stage faces the big N problem giving rise to complex memory and computational problems. As a solution to the big N problem, we propose a Gaussian Markov random field (GMRF) for innovations with the Matérn covariance matrix obtained from the corresponding Gaussian field (GF) matrix by means of the stochastic partial differential equation (SPDE) method and the finite element method (FEM). For inference, we propose Bayesian statistics and integrated nested Laplace approximation (INLA) in the R-INLA package. The above approach is demonstrated using daily data collected from 13 PM2.5 monitoring stations in Jakarta Province, Indonesia, for 1 January–31 December 2022. The first stage of the model generates PM2.5 forecasts for the 13 monitoring stations for the period 1–31 January 2023, imputing missing data by means of the MSTS model. To capture temporal trends in the PM2.5 concentrations, the model applies a first-order autoregressive process and a seasonal process. The second stage involves creating a high-resolution map for the period 1–31 January 2023, for sampled and non-sampled spatiotemporal units. It uses the MSTS-generated PM2.5 predictions for the sampled spatiotemporal units and observations of the covariate’s altitude, population density, and rainfall for sampled and non-samples spatiotemporal units. For the spatially correlated random effects, we apply a first-order random walk process. The validation of out-of-sample forecasts indicates a strong model fit with low mean squared error (0.001), mean absolute error (0.037), and mean absolute percentage error (0.041), and a high R² value (0.855). The analysis reveals that altitude and precipitation negatively impact PM2.5 concentrations, while population density has a positive effect. Specifically, a one-meter increase in altitude is linked to a 7.8% decrease in PM2.5, while a one-person increase in population density leads to a 7.0% rise in PM2.5. Additionally, a one-millimeter increase in rainfall corresponds to a 3.9% decrease in PM2.5. The paper makes a valuable contribution to the field of forecasting high-resolution PM2.5 levels, which is essential for providing detailed, accurate information for public health policy. The approach presents a new and innovative method for addressi
{"title":"High-Resolution Spatiotemporal Forecasting with Missing Observations Including an Application to Daily Particulate Matter 2.5 Concentrations in Jakarta Province, Indonesia","authors":"I Gede Nyoman Mindra Jaya, Henk Folmer","doi":"10.3390/math12182899","DOIUrl":"https://doi.org/10.3390/math12182899","url":null,"abstract":"Accurate forecasting of high-resolution particulate matter 2.5 (PM2.5) levels is essential for the development of public health policy. However, datasets used for this purpose often contain missing observations. This study presents a two-stage approach to handle this problem. The first stage is a multivariate spatial time series (MSTS) model, used to generate forecasts for the sampled spatial units and to impute missing observations. The MSTS model utilizes the similarities between the temporal patterns of the time series of the spatial units to impute the missing data across space. The second stage is the high-resolution prediction model, which generates predictions that cover the entire study domain. The second stage faces the big N problem giving rise to complex memory and computational problems. As a solution to the big N problem, we propose a Gaussian Markov random field (GMRF) for innovations with the Matérn covariance matrix obtained from the corresponding Gaussian field (GF) matrix by means of the stochastic partial differential equation (SPDE) method and the finite element method (FEM). For inference, we propose Bayesian statistics and integrated nested Laplace approximation (INLA) in the R-INLA package. The above approach is demonstrated using daily data collected from 13 PM2.5 monitoring stations in Jakarta Province, Indonesia, for 1 January–31 December 2022. The first stage of the model generates PM2.5 forecasts for the 13 monitoring stations for the period 1–31 January 2023, imputing missing data by means of the MSTS model. To capture temporal trends in the PM2.5 concentrations, the model applies a first-order autoregressive process and a seasonal process. The second stage involves creating a high-resolution map for the period 1–31 January 2023, for sampled and non-sampled spatiotemporal units. It uses the MSTS-generated PM2.5 predictions for the sampled spatiotemporal units and observations of the covariate’s altitude, population density, and rainfall for sampled and non-samples spatiotemporal units. For the spatially correlated random effects, we apply a first-order random walk process. The validation of out-of-sample forecasts indicates a strong model fit with low mean squared error (0.001), mean absolute error (0.037), and mean absolute percentage error (0.041), and a high R² value (0.855). The analysis reveals that altitude and precipitation negatively impact PM2.5 concentrations, while population density has a positive effect. Specifically, a one-meter increase in altitude is linked to a 7.8% decrease in PM2.5, while a one-person increase in population density leads to a 7.0% rise in PM2.5. Additionally, a one-millimeter increase in rainfall corresponds to a 3.9% decrease in PM2.5. The paper makes a valuable contribution to the field of forecasting high-resolution PM2.5 levels, which is essential for providing detailed, accurate information for public health policy. The approach presents a new and innovative method for addressi","PeriodicalId":18303,"journal":{"name":"Mathematics","volume":"1 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249110","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}
Autism spectrum disorder (ASD) is a complex neurodevelopmental condition that presents significant diagnostic challenges due to its varied symptoms and nature. This study aims to improve ASD classification using advanced deep learning techniques applied to neuroimaging data. We developed an automated system leveraging the ABIDE-1 dataset and a novel lightweight quantized one-dimensional (1D) Convolutional Neural Network (Q-CNN) model to analyze fMRI data. Our approach employs the NIAK pipeline with multiple brain atlases and filtering methods. Initially, the Regions of Interest (ROIs) are converted into feature vectors using tangent space embedding to feed into the Q-CNN model. The proposed 1D-CNN is quantized through Quantize Aware Training (QAT). As the quantization method, int8 quantization is utilized, which makes it both robust and lightweight. We propose a federated learning (FL) framework to ensure data privacy, which allows decentralized training across different data centers without compromising local data security. Our findings indicate that the CC200 brain atlas, within the NIAK pipeline’s filt-global filtering methods, provides the best results for ASD classification. Notably, the ASD classification outcomes have achieved a significant test accuracy of 98% using the CC200 and filt-global filtering techniques. To the best of our knowledge, this performance surpasses previous studies in the field, highlighting a notable enhancement in ASD detection from fMRI data. Furthermore, the FL-based Q-CNN model demonstrated robust performance and high efficiency on a Raspberry Pi 4, underscoring its potential for real-world applications. We exhibit the efficacy of the Q-CNN model by comparing its inference time, power consumption, and storage requirements with those of the 1D-CNN, quantized CNN, and the proposed int8 Q-CNN models. This research has made several key contributions, including the development of a lightweight int8 Q-CNN model, the application of FL for data privacy, and the evaluation of the proposed model in real-world settings. By identifying optimal brain atlases and filtering methods, this study provides valuable insights for future research in the field of neurodevelopmental disorders.
{"title":"Enhancing Autism Spectrum Disorder Classification with Lightweight Quantized CNNs and Federated Learning on ABIDE-1 Dataset","authors":"Simran Gupta, Md. Rahad Islam Bhuiyan, Sadia Sultana Chowa, Sidratul Montaha, Rashik Rahman, Sk. Tanzir Mehedi, Ziaur Rahman","doi":"10.3390/math12182886","DOIUrl":"https://doi.org/10.3390/math12182886","url":null,"abstract":"Autism spectrum disorder (ASD) is a complex neurodevelopmental condition that presents significant diagnostic challenges due to its varied symptoms and nature. This study aims to improve ASD classification using advanced deep learning techniques applied to neuroimaging data. We developed an automated system leveraging the ABIDE-1 dataset and a novel lightweight quantized one-dimensional (1D) Convolutional Neural Network (Q-CNN) model to analyze fMRI data. Our approach employs the NIAK pipeline with multiple brain atlases and filtering methods. Initially, the Regions of Interest (ROIs) are converted into feature vectors using tangent space embedding to feed into the Q-CNN model. The proposed 1D-CNN is quantized through Quantize Aware Training (QAT). As the quantization method, int8 quantization is utilized, which makes it both robust and lightweight. We propose a federated learning (FL) framework to ensure data privacy, which allows decentralized training across different data centers without compromising local data security. Our findings indicate that the CC200 brain atlas, within the NIAK pipeline’s filt-global filtering methods, provides the best results for ASD classification. Notably, the ASD classification outcomes have achieved a significant test accuracy of 98% using the CC200 and filt-global filtering techniques. To the best of our knowledge, this performance surpasses previous studies in the field, highlighting a notable enhancement in ASD detection from fMRI data. Furthermore, the FL-based Q-CNN model demonstrated robust performance and high efficiency on a Raspberry Pi 4, underscoring its potential for real-world applications. We exhibit the efficacy of the Q-CNN model by comparing its inference time, power consumption, and storage requirements with those of the 1D-CNN, quantized CNN, and the proposed int8 Q-CNN models. This research has made several key contributions, including the development of a lightweight int8 Q-CNN model, the application of FL for data privacy, and the evaluation of the proposed model in real-world settings. By identifying optimal brain atlases and filtering methods, this study provides valuable insights for future research in the field of neurodevelopmental disorders.","PeriodicalId":18303,"journal":{"name":"Mathematics","volume":"100 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249112","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 paper, we first prove the existence and uniqueness of the solution to a variable-order Caputo–Fabrizio fractional stochastic differential equation driven by a multiplicative white noise, which describes random phenomena with non-local effects and non-singular kernels. The Euler–Maruyama scheme is extended to develop the Euler–Maruyama method, and the strong convergence of the proposed method is demonstrated. The main difference between our work and the existing literature is the fact that our assumptions on the nonlinear external forces are those of one-sided Lipschitz conditions on both the drift and the nonlinear intensity of the noise as well as the proofs of the higher integrability of the solution and the approximating sequence. Finally, to validate the numerical approach, current results from the numerical implementation are presented to test the efficiency of the scheme used in order to substantiate the theoretical analysis.
{"title":"Strong Convergence of Euler-Type Methods for Nonlinear Fractional Stochastic Differential Equations without Singular Kernel","authors":"Zakaria Ali, Minyahil Abera Abebe, Talat Nazir","doi":"10.3390/math12182890","DOIUrl":"https://doi.org/10.3390/math12182890","url":null,"abstract":"In this paper, we first prove the existence and uniqueness of the solution to a variable-order Caputo–Fabrizio fractional stochastic differential equation driven by a multiplicative white noise, which describes random phenomena with non-local effects and non-singular kernels. The Euler–Maruyama scheme is extended to develop the Euler–Maruyama method, and the strong convergence of the proposed method is demonstrated. The main difference between our work and the existing literature is the fact that our assumptions on the nonlinear external forces are those of one-sided Lipschitz conditions on both the drift and the nonlinear intensity of the noise as well as the proofs of the higher integrability of the solution and the approximating sequence. Finally, to validate the numerical approach, current results from the numerical implementation are presented to test the efficiency of the scheme used in order to substantiate the theoretical analysis.","PeriodicalId":18303,"journal":{"name":"Mathematics","volume":"51 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249133","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}
We establish a relationship between stochastic differential games (SDGs) and a unified forward–backward coupled stochastic partial differential equation (SPDE) with discontinuous Lévy Jumps. The SDGs have q players and are driven by a general-dimensional vector Lévy process. By establishing a vector-form Ito-Ventzell formula and a 4-tuple vector-field solution to the unified SPDE, we obtain a Pareto optimal Nash equilibrium policy process or a saddle point policy process to the SDG in a non-zero-sum or zero-sum sense. The unified SPDE is in both a general-dimensional vector form and forward–backward coupling manner. The partial differential operators in its drift, diffusion, and jump coefficients are in time-variable and position parameters over a domain. Since the unified SPDE is of general nonlinearity and a general high order, we extend our recent study from the existing Brownian motion (BM)-driven backward case to a general Lévy-driven forward–backward coupled case. In doing so, we construct a new topological space to support the proof of the existence and uniqueness of an adapted solution of the unified SPDE, which is in a 4-tuple strong sense. The construction of the topological space is through constructing a set of topological spaces associated with a set of exponents {γ1,γ2,…} under a set of general localized conditions, which is significantly different from the construction of the single exponent case. Furthermore, due to the coupling from the forward SPDE and the involvement of the discontinuous Lévy jumps, our study is also significantly different from the BM-driven backward case. The coupling between forward and backward SPDEs essentially corresponds to the interaction between noise encoding and noise decoding in the current hot diffusion transformer model for generative AI.
{"title":"Stochastic Differential Games and a Unified Forward–Backward Coupled Stochastic Partial Differential Equation with Lévy Jumps","authors":"Wanyang Dai","doi":"10.3390/math12182891","DOIUrl":"https://doi.org/10.3390/math12182891","url":null,"abstract":"We establish a relationship between stochastic differential games (SDGs) and a unified forward–backward coupled stochastic partial differential equation (SPDE) with discontinuous Lévy Jumps. The SDGs have q players and are driven by a general-dimensional vector Lévy process. By establishing a vector-form Ito-Ventzell formula and a 4-tuple vector-field solution to the unified SPDE, we obtain a Pareto optimal Nash equilibrium policy process or a saddle point policy process to the SDG in a non-zero-sum or zero-sum sense. The unified SPDE is in both a general-dimensional vector form and forward–backward coupling manner. The partial differential operators in its drift, diffusion, and jump coefficients are in time-variable and position parameters over a domain. Since the unified SPDE is of general nonlinearity and a general high order, we extend our recent study from the existing Brownian motion (BM)-driven backward case to a general Lévy-driven forward–backward coupled case. In doing so, we construct a new topological space to support the proof of the existence and uniqueness of an adapted solution of the unified SPDE, which is in a 4-tuple strong sense. The construction of the topological space is through constructing a set of topological spaces associated with a set of exponents {γ1,γ2,…} under a set of general localized conditions, which is significantly different from the construction of the single exponent case. Furthermore, due to the coupling from the forward SPDE and the involvement of the discontinuous Lévy jumps, our study is also significantly different from the BM-driven backward case. The coupling between forward and backward SPDEs essentially corresponds to the interaction between noise encoding and noise decoding in the current hot diffusion transformer model for generative AI.","PeriodicalId":18303,"journal":{"name":"Mathematics","volume":"2 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249135","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}
This paper investigates the propagation of in-plane surface waves in a coated thermoelastic half-space. First, it investigates a special case where the surface layer is described by the Maxwell–Cattaneo thermoelastic approach, while the half-space is filled by a thermoelastic material described by the classical Fourier law for the heat flux. The contact between the layer and the half-space is assumed to be welded, i.e., the displacements and the temperature, as well as the stresses and the heat flux are continuous through the interface of the layer and the half-space. The boundary and continuity conditions of the problem are formulated and then the exact dispersion relation of the surface waves is established. An illustrative numerical simulation is presented for the case of an aluminum thermoelastic layer coating a thermoelastic copper half-space, highlighting important aspects regarding the propagation of Rayleigh waves in such structures. The exact effective boundary conditions at the interface are also established replacing the entire effect of the layer on the half-space. The general case of the problem is also investigated when both the surface layer and the half-space are described by the Maxwell–Cattaneo thermoelasticity theory. This study helps to further understand the propagation characteristics of elastic waves in layered structures with thermal effects described by the Maxwell–Cattaneo approach.
{"title":"Rayleigh Waves in a Thermoelastic Half-Space Coated by a Maxwell–Cattaneo Thermoelastic Layer","authors":"Stan Chiriţă, Ciro D’Apice","doi":"10.3390/math12182885","DOIUrl":"https://doi.org/10.3390/math12182885","url":null,"abstract":"This paper investigates the propagation of in-plane surface waves in a coated thermoelastic half-space. First, it investigates a special case where the surface layer is described by the Maxwell–Cattaneo thermoelastic approach, while the half-space is filled by a thermoelastic material described by the classical Fourier law for the heat flux. The contact between the layer and the half-space is assumed to be welded, i.e., the displacements and the temperature, as well as the stresses and the heat flux are continuous through the interface of the layer and the half-space. The boundary and continuity conditions of the problem are formulated and then the exact dispersion relation of the surface waves is established. An illustrative numerical simulation is presented for the case of an aluminum thermoelastic layer coating a thermoelastic copper half-space, highlighting important aspects regarding the propagation of Rayleigh waves in such structures. The exact effective boundary conditions at the interface are also established replacing the entire effect of the layer on the half-space. The general case of the problem is also investigated when both the surface layer and the half-space are described by the Maxwell–Cattaneo thermoelasticity theory. This study helps to further understand the propagation characteristics of elastic waves in layered structures with thermal effects described by the Maxwell–Cattaneo approach.","PeriodicalId":18303,"journal":{"name":"Mathematics","volume":"49 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249111","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}
Sequential recommendations aim to predict users’ next interactions by modeling their interaction sequences. Most existing work concentrates on user preferences within these sequences, overlooking the complex item relationships across sequences. Additionally, these studies often fail to address the diversity of user interests, thus not capturing their varied latent preferences effectively. To tackle these problems, this paper develops a novel recommendation algorithm based on hypergraphs and contrastive learning named HyperCLR. It dynamically incorporates the time and location embeddings of items to model high-order relationships in user preferences. Moreover, we developed a graph construction approach named IFDG, which utilizes global item visit frequencies and spatial distances to discern item relevancy. By sampling subgraphs from IFDG, HyperCLR can align the representations of identical interaction sequences closely while distinguishing them from the broader global context on IFDG. This approach enhances the accuracy of sequential recommendations. Furthermore, a gating mechanism is designed to tailor the global context information to individual user preferences. Extensive experiments on Taobao, Books and Games datasets have shown that HyperCLR consistently surpasses baselines, demonstrating the effectiveness of the method. In particular, in comparison to the best baseline methods, HyperCLR demonstrated a 29.1% improvement in performance on the Taobao dataset.
{"title":"HyperCLR: A Personalized Sequential Recommendation Algorithm Based on Hypergraph and Contrastive Learning","authors":"Ruiqi Zhang, Haitao Wang, Jianfeng He","doi":"10.3390/math12182887","DOIUrl":"https://doi.org/10.3390/math12182887","url":null,"abstract":"Sequential recommendations aim to predict users’ next interactions by modeling their interaction sequences. Most existing work concentrates on user preferences within these sequences, overlooking the complex item relationships across sequences. Additionally, these studies often fail to address the diversity of user interests, thus not capturing their varied latent preferences effectively. To tackle these problems, this paper develops a novel recommendation algorithm based on hypergraphs and contrastive learning named HyperCLR. It dynamically incorporates the time and location embeddings of items to model high-order relationships in user preferences. Moreover, we developed a graph construction approach named IFDG, which utilizes global item visit frequencies and spatial distances to discern item relevancy. By sampling subgraphs from IFDG, HyperCLR can align the representations of identical interaction sequences closely while distinguishing them from the broader global context on IFDG. This approach enhances the accuracy of sequential recommendations. Furthermore, a gating mechanism is designed to tailor the global context information to individual user preferences. Extensive experiments on Taobao, Books and Games datasets have shown that HyperCLR consistently surpasses baselines, demonstrating the effectiveness of the method. In particular, in comparison to the best baseline methods, HyperCLR demonstrated a 29.1% improvement in performance on the Taobao dataset.","PeriodicalId":18303,"journal":{"name":"Mathematics","volume":"53 1","pages":""},"PeriodicalIF":2.4,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249130","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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