Mathematics

22 pages, 783 KiB

Open AccessArticle

Learned Query Optimization by Constraint-Based Query Plan Augmentation

by Chen Ye, Haoyang Duan, Hua Zhang, Yifan Wu and Guojun Dai

Mathematics 2024, 12(19), 3102; https://doi.org/10.3390/math12193102 - 3 Oct 2024

Over the last decades, various cost-based optimizers have been proposed to generate optimal plans for SQL queries. These optimizers are key to achieving good performance in database systems and can speed up query execution. Still, they may need enormous expert efforts and perform [...] Read more.

Over the last decades, various cost-based optimizers have been proposed to generate optimal plans for SQL queries. These optimizers are key to achieving good performance in database systems and can speed up query execution. Still, they may need enormous expert efforts and perform poorly on complicated queries. Learning-based optimizers have been shown to achieve high-quality plans by learning from past experiences. However, these solutions treat each query separately and neglect the semantic equivalence among different queries. Intuitively, a high-quality plan may be obtained for a complicated query by discovering a simple equivalent query. Motivated by this, in this paper, we present Celo, a novel constraint-enhanced learned optimizer to directly integrate the equivalent information of queries into the learning-based model. We apply denial constraints to identify equivalent queries by replacing equivalent predicates. Given a query, we augment the query plans generated by the learning-based model with the high-quality plans of its equivalent queries. Then, a more potentially well-performed plan will be predicted among the augmented query plans. Extensive experiments using real-world datasets demonstrated that Celo outperforms the previous state-of-the-art (SOTA) results even with few constraints. Full article

(This article belongs to the Section Mathematics and Computer Science)

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16 pages, 1027 KiB

Open AccessFeature PaperArticle

Tight 9-Cycle Decompositions of λ-Fold Complete 3-Uniform Hypergraphs

by Hongtao Zhao and Jianxiao Gu

Mathematics 2024, 12(19), 3101; https://doi.org/10.3390/math12193101 (registering DOI) - 3 Oct 2024

Abstract

For

2 \leq t \leq m

, let

Z_{m}

denote the group of integers modulo m, and let

T C_{m}^{(t)}

denote the t-uniform hypergraph with vertex set

Z_{m}

and hyperedge set [...] Read more.

For

2 \leq t \leq m

, let

Z_{m}

denote the group of integers modulo m, and let

T C_{m}^{(t)}

denote the t-uniform hypergraph with vertex set

Z_{m}

and hyperedge set

{{i, i + 1, i + 2, \dots, i + t - 1} : i \in Z_{m}}

. Any hypergraph isomorphic to

T C_{m}^{(t)}

is a t-uniform tight m-cycle. In this paper, we consider the existence of tight 9-cycle decompositions of

λ

-fold complete 3-uniform hypergraphs. According to the recursive constructions, the required designs of small orders are found. For hypergraphs with large orders, they can be recursively generated using some designs of small orders. Then, we obtain the necessary and sufficient conditions for the existence of

T C_{9}^{(3)}

-decomposition of

λ K_{n}^{(3)}

. We show there exists a

T C_{9}^{(3)}

-decomposition of

λ K_{n}^{(3)}

if and only if

λ n (n - 1) (n - 2) \equiv 0 (mod 54)

,

λ (n - 1) (n - 2) \equiv 0 (mod 6)

and

n \geq 9

. Full article

18 pages, 1266 KiB

Open AccessFeature PaperArticle

Finite-Time Mass Estimation Using ℋ_∞ and Sliding Mode Control for a Multicopter

by Carlos Augusto Arellano-Muro, Guillermo Luis Osuna-González and Riccardo Cespi

Mathematics 2024, 12(19), 3100; https://doi.org/10.3390/math12193100 - 3 Oct 2024

Abstract

Nonlinear control theory applied to unmanned aeronautical vehicles is an engineering topic that has received higher and higher popularity during the last decade. Model-based control approaches have shown increased performance in flight control accuracy and robustness compared to model-free proposals based on parameter [...] Read more.

Nonlinear control theory applied to unmanned aeronautical vehicles is an engineering topic that has received higher and higher popularity during the last decade. Model-based control approaches have shown increased performance in flight control accuracy and robustness compared to model-free proposals based on parameter adaptation and estimation. However, model-based structures need more computational efforts in terms of spatial and temporal variables. To avoid these constraints, the latest drone flight controls are based on quaternion models, ensuring more advanced computational performances. To this aim, this paper deals with a flight control algorithm of a quadrotor, in which the mathematics model of the plant is defined in terms of quaternions. Additionally, when aerial vehicles are used in specific applications such as slung load transportation and agriculture fields, among others, the variation of the mass receives high importance since it could make the entire system unstable. In the same line of ideas, this paper presents a

H_{\infty}

strategy, combined with a Super-Twisting Sliding-Mode Control, ensuring the control objective of the mass variations identification, and trajectory tracking, to be solved. The stability analysis of the proposed control approach is also discussed, and the quality and performances of the presented control strategy are tested by simulations, in an interesting case in which mass variations and external perturbations cannot be negligible. Full article

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13 pages, 1088 KiB

Open AccessArticle

Generalized Kelvin–Voigt Creep Model in Fractal Space–Time

by Eduardo Reyes de Luna, Andriy Kryvko, Juan B. Pascual-Francisco, Ignacio Hernández and Didier Samayoa

Mathematics 2024, 12(19), 3099; https://doi.org/10.3390/math12193099 - 3 Oct 2024

Abstract

In this paper, we study the creep phenomena for self-similar models of viscoelastic materials and derive a generalization of the Kelvin–Voigt model in the framework of fractal continuum calculus. Creep compliance for the Kelvin–Voigt model is extended to fractal manifolds through local fractal-continuum [...] Read more.

In this paper, we study the creep phenomena for self-similar models of viscoelastic materials and derive a generalization of the Kelvin–Voigt model in the framework of fractal continuum calculus. Creep compliance for the Kelvin–Voigt model is extended to fractal manifolds through local fractal-continuum differential operators. Generalized fractal creep compliance is obtained, taking into account the intrinsic time

τ

and the fractal dimension of time-scale

β

. The model obtained is validated with experimental data obtained for resin samples with the fractal structure of a Sierpinski carpet and experimental data on rock salt. Comparisons of the model predictions with the experimental data are presented as the curves of slow continuous deformations. Full article

(This article belongs to the Special Issue The Applications of Fractional Calculus in Control Engineering, Dynamical Systems and Signal Processing)

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41 pages, 2413 KiB

Open AccessArticle

Q-Learning-Based Dumbo Octopus Algorithm for Parameter Tuning of Fractional-Order PID Controller for AVR Systems

by Yuanyuan Li, Lei Ni, Geng Wang, Sumeet S. Aphale and Lanqiang Zhang

Mathematics 2024, 12(19), 3098; https://doi.org/10.3390/math12193098 - 3 Oct 2024

Abstract

The tuning of fractional-order proportional-integral-derivative (FOPID) controllers for automatic voltage regulator (AVR) systems presents a complex challenge, necessitating the solution of real-order integral and differential equations. This study introduces the Dumbo Octopus Algorithm (DOA), a novel metaheuristic inspired by machine learning with animal [...] Read more.

The tuning of fractional-order proportional-integral-derivative (FOPID) controllers for automatic voltage regulator (AVR) systems presents a complex challenge, necessitating the solution of real-order integral and differential equations. This study introduces the Dumbo Octopus Algorithm (DOA), a novel metaheuristic inspired by machine learning with animal behaviors, as an innovative approach to address this issue. For the first time, the DOA is invented and employed to optimize FOPID parameters, and its performance is rigorously evaluated against 11 existing metaheuristic algorithms using 23 classical benchmark functions and CEC2019 test sets. The evaluation includes a comprehensive quantitative analysis and qualitative analysis. Statistical significance was assessed using the Friedman’s test, highlighting the superior performance of the DOA compared to competing algorithms. To further validate its effectiveness, the DOA was applied to the FOPID parameter tuning of an AVR system, demonstrating exceptional performance in practical engineering applications. The results indicate that the DOA outperforms other algorithms in terms of convergence accuracy, robustness, and practical problem-solving capability. This establishes the DOA as a superior and promising solution for complex optimization problems, offering significant advancements in the tuning of FOPID for AVR systems. Full article

(This article belongs to the Special Issue Advanced Computational Intelligence)

22 pages, 312 KiB

Open AccessArticle

Existence and Stability Results for Thermodiffusion Laminated Beam System with Delay Feedback

by Zineb Khalili, Djamel Ouchenane, Ali Krelifa, Imene Laribi, Salah Boulaaras and Ahmed Himadan Ahmed

Mathematics 2024, 12(19), 3097; https://doi.org/10.3390/math12193097 - 3 Oct 2024

Abstract

In this paper, a one-dimensional thermodiffusion laminated beam system with delay feedback is studied. The existence of a solution for our system is discussed within the context of the semigroup approach. In addition, under different boundary conditions, two results of stability properties independent [...] Read more.

In this paper, a one-dimensional thermodiffusion laminated beam system with delay feedback is studied. The existence of a solution for our system is discussed within the context of the semigroup approach. In addition, under different boundary conditions, two results of stability properties independent of initial data are investigated. Full article

22 pages, 3944 KiB

Open AccessFeature PaperArticle

Analytical C² Continuous Surface Blending

by Xiangyu You, Feng Tian, Wen Tang, Jian Chang and Jianjun Zhang

Mathematics 2024, 12(19), 3096; https://doi.org/10.3390/math12193096 - 3 Oct 2024

Abstract

Surface blending is an important topic in geometric modelling and is widely applied in computer-aided design and creative industries to create smooth transition surfaces. Among various surface blending methods, partial differential equation (PDE)-based surface blending has the advantages of effective shape control and [...] Read more.

Surface blending is an important topic in geometric modelling and is widely applied in computer-aided design and creative industries to create smooth transition surfaces. Among various surface blending methods, partial differential equation (PDE)-based surface blending has the advantages of effective shape control and exact satisfaction of blending boundary constraints. However, it is not easy to solve partial differential equations subjected to blending boundary constraints. In this paper, we investigate how to solve PDEs analytically and develop an analytical PDE-based method to achieve surface blending with

C^{2}

continuity. Taking advantage of elementary functions identified from blending boundary constraints, our proposed method first changes blending boundary constraints into a linear combination of the identified elementary functions. Accordingly, the functions for blending surfaces are constructed from these elementary functions, which transform sixth-order partial differential equations for

C^{2}

surface blending into sixth-order ordinary differential equations (ODEs). We investigate the analytical solutions of the transformed sixth-order ordinary differential equations subjected to corresponding blending boundary constraints. With the developed analytical PDE-based method, we solve

C^{2}

continuous surface blending problems. The surface blending example presented in this paper indicates that the developed method is simple and easy to use. It can be used to effectively control the shape of blending surfaces and at the same time exactly satisfy

C^{2}

continuous blending boundary constraints. Full article

(This article belongs to the Special Issue Mathematical Applications in Computer Graphics)

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12 pages, 266 KiB

Open AccessArticle

Introducing the Third-Order Fuzzy Superordination Concept and Related Results

by Georgia Irina Oros, Simona Dzitac and Daniela Andrada Bardac-Vlada

Mathematics 2024, 12(19), 3095; https://doi.org/10.3390/math12193095 - 2 Oct 2024

Abstract

Third-order fuzzy differential subordination studies were recently initiated by developing the main concepts necessary for obtaining new results on this topic. The present paper introduces the dual concept of third-order fuzzy differential superordination by building on the known results that are valid for [...] Read more.

Third-order fuzzy differential subordination studies were recently initiated by developing the main concepts necessary for obtaining new results on this topic. The present paper introduces the dual concept of third-order fuzzy differential superordination by building on the known results that are valid for second-order fuzzy differential superordination. The outcome of this study offers necessary and sufficient conditions for determining subordinants of a third-order fuzzy differential superordination and, furthermore, for finding the best subordinant for such fuzzy differential superordiantion, when it can be obtained. An example to suggest further uses of the new outcome reported in this work is enclosed to conclude this study. Full article

(This article belongs to the Special Issue Advanced Research in Complex Analysis Operators and Special Classes of Analytic Functions)

20 pages, 574 KiB

Open AccessFeature PaperArticle

Mittag–Leffler Fractional Stochastic Integrals and Processes with Applications

by Enrica Pirozzi

Mathematics 2024, 12(19), 3094; https://doi.org/10.3390/math12193094 - 2 Oct 2024

Abstract

We study Mittag–Leffler (ML) fractional integrals involved in the solution processes of a system of coupled fractional stochastic differential equations. We introduce the ML fractional stochastic process as a ML fractional stochastic integral with respect to a standard Brownian motion. We provide some [...] Read more.

We study Mittag–Leffler (ML) fractional integrals involved in the solution processes of a system of coupled fractional stochastic differential equations. We introduce the ML fractional stochastic process as a ML fractional stochastic integral with respect to a standard Brownian motion. We provide some representation formulas of solution processes in terms of Mittag–Leffler fractional integrals and processes. Computable expressions of the mean functions and of the covariances of such processes are specifically given. The application in neuronal modeling is provided, and all involved functions and processes are specifically determined. Numerical evaluations are carried out and some results are shown and discussed. Full article

(This article belongs to the Special Issue Stochastic Processes: Theory, Simulation and Applications)

18 pages, 11050 KiB

Open AccessArticle

Mitigating Adversarial Attacks in Object Detection through Conditional Diffusion Models

by Xudong Ye, Qi Zhang, Sanshuai Cui, Zuobin Ying, Jingzhang Sun and Xia Du

Mathematics 2024, 12(19), 3093; https://doi.org/10.3390/math12193093 - 2 Oct 2024

Abstract

The field of object detection has witnessed significant advancements in recent years, thanks to the remarkable progress in artificial intelligence and deep learning. These breakthroughs have significantly enhanced the accuracy and efficiency of detecting and categorizing objects in digital images. Nonetheless, contemporary object [...] Read more.

The field of object detection has witnessed significant advancements in recent years, thanks to the remarkable progress in artificial intelligence and deep learning. These breakthroughs have significantly enhanced the accuracy and efficiency of detecting and categorizing objects in digital images. Nonetheless, contemporary object detection technologies have certain limitations, such as their inability to counter white-box attacks, insufficient denoising, suboptimal reconstruction, and gradient confusion. To overcome these hurdles, this study proposes an innovative approach that uses conditional diffusion models to perturb adversarial examples. The process begins with the application of a random chessboard mask to the adversarial example, followed by the addition of a slight noise to fill the masked area during the forward process. The adversarial image is then restored to its original form through a reverse generative process that only considers the masked pixels, not the entire image. Next, we use the complement of the initial mask as the mask for the second stage to reconstruct the image once more. This two-stage masking process allows for the complete removal of global disturbances and aids in image reconstruction. In particular, we employ a conditional diffusion model based on a class-conditional U-Net architecture, with the source image further conditioned through concatenation. Our method outperforms the recently introduced HARP method by 5% and 6.5% in mAP on the COCO2017 and PASCAL VOC datasets, respectively, under non-APT PGD attacks. Comprehensive experimental results confirm that our method can effectively restore adversarial examples, demonstrating its practical utility. Full article

(This article belongs to the Special Issue Mathematics-Based Methods in Artificial Intelligence, Pattern Recognition and Deep Learning, 2nd Edition)

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19 pages, 1342 KiB

Open AccessArticle

Anomaly Detection over Streaming Graphs with Finger-Based Higher-Order Graph Sketch

by Min Lu, Qianzhen Zhang and Xianqiang Zhu

Mathematics 2024, 12(19), 3092; https://doi.org/10.3390/math12193092 - 2 Oct 2024

Abstract

A streaming graph is a constantly growing sequence of edges, which forms a dynamic graph that changes with every edge in the stream. An anomalous behavior in a streaming graph can be modeled as an edge or a subgraph that is unusual compared [...] Read more.

A streaming graph is a constantly growing sequence of edges, which forms a dynamic graph that changes with every edge in the stream. An anomalous behavior in a streaming graph can be modeled as an edge or a subgraph that is unusual compared to the rest of the graph. Identifying anomalous behaviors in real time is essential to the early warning of abnormal or notable events. Due to the complexity of the problem, little work has been reported so far to solve the problem. In this paper, we propose Finger-based Higher-order Graph Sketch (FHGS for short), which is an approximate data structure for streaming graphs with linear memory usage, high update speed, and high accuracy and supports both edge and subgraph anomaly detection. FHGS first maps each edge into a matrix based on hash functions, and then counts its frequency in a time window with unique fingerprints for detecting anomalies. Extensive experiments confirm that our approach generate high-quality results compared to baseline methods. Full article

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19 pages, 277 KiB

Open AccessArticle

An Exploration on Z-Number and Its Properties

by Guanshuang Jiang, Xinyu Shen, Xuefei Liao, Xiaoqi Xuan, Lechen Wu, Haomin Zhang and Zhen Li

Mathematics 2024, 12(19), 3091; https://doi.org/10.3390/math12193091 - 2 Oct 2024

Abstract

The Z-number deserves further exploration in uncertain environments to effectively address fuzziness and reliability in practical issues simultaneously. Based on the credibility distribution and conversion rules, we define its expected value, variance, and semi-variance, and demonstrate the feasibility of these calculations by deriving [...] Read more.

The Z-number deserves further exploration in uncertain environments to effectively address fuzziness and reliability in practical issues simultaneously. Based on the credibility distribution and conversion rules, we define its expected value, variance, and semi-variance, and demonstrate the feasibility of these calculations by deriving formulas. Moreover, we explore three characteristics inherent in symmetrical Z-numbers. The link between the variance and semi-variance of Z-numbers is discovered and proved. Furthermore, we apply the formulas for expected value and variance to Z-numbers in examples, the results of which validate our proposed formulas. The findings underscore the significance of our study in applying the expected value and variance of fuzzy sets across diverse fields. Full article

14 pages, 1126 KiB

Open AccessArticle

Adaptive Bi-Encoder Model Selection and Ensemble for Text Classification

by Youngki Park and Youhyun Shin

Mathematics 2024, 12(19), 3090; https://doi.org/10.3390/math12193090 - 2 Oct 2024

Abstract

Can bi-encoders, without additional fine-tuning, achieve a performance comparable to fine-tuned BERT models in classification tasks? To answer this question, we present a simple yet effective approach to text classification using bi-encoders without the need for fine-tuning. Our main observation is that state-of-the-art [...] Read more.

Can bi-encoders, without additional fine-tuning, achieve a performance comparable to fine-tuned BERT models in classification tasks? To answer this question, we present a simple yet effective approach to text classification using bi-encoders without the need for fine-tuning. Our main observation is that state-of-the-art bi-encoders exhibit varying performance across different datasets. Therefore, our proposed approaches involve preparing multiple bi-encoders and, when a new dataset is provided, selecting and ensembling the most appropriate ones based on the dataset. Experimental results show that, for text classification tasks on subsets of the AG News, SMS Spam Collection, Stanford Sentiment Treebank v2, and TREC Question Classification datasets, the proposed approaches achieve performance comparable to fine-tuned BERT-Base, DistilBERT-Base, ALBERT-Base, and RoBERTa-Base. For instance, using the well-known bi-encoder model all-MiniLM-L12-v2 without additional optimization resulted in an average accuracy of 77.84%. This improved to 89.49% through the application of the proposed adaptive selection and ensemble techniques, and further increased to 91.96% when combined with the RoBERTa-Base model. We believe that this approach will be particularly useful in fields such as K-12 AI programming education, where pre-trained models are applied to small datasets without fine-tuning. Full article

(This article belongs to the Special Issue New Machine Learning and Deep Learning Techniques in Natural Language Processing)

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24 pages, 751 KiB

Open AccessArticle

Applying the Simulated Annealing Algorithm to the Set Orienteering Problem with Mandatory Visits

by Shih-Wei Lin, Si-Rui Guo and Wen-Jie Wu

Mathematics 2024, 12(19), 3089; https://doi.org/10.3390/math12193089 - 2 Oct 2024

Abstract

This study addresses the set orienteering problem with mandatory visits (SOPMV), a variant of the team orienteering problem (SOP). In SOPMV, certain critical sets must be visited. The study began by formulating the mathematical model for SOPMV. To tackle the challenge of obtaining [...] Read more.

This study addresses the set orienteering problem with mandatory visits (SOPMV), a variant of the team orienteering problem (SOP). In SOPMV, certain critical sets must be visited. The study began by formulating the mathematical model for SOPMV. To tackle the challenge of obtaining a feasible route within time constraints using the original MILP approach, a two-stage mixed-integer linear programming (MILP) model is proposed. Subsequently, a simulated annealing (SA) algorithm and a dynamic programming method were employed to identify the optimal route. The proposed SA algorithm was used to solve the SOP and was compared to other algorithms, demonstrating its effectiveness. The SA was then applied to solve the SOPMV problem. The results indicate that the solutions obtained using SA are superior and more efficient compared to those derived from the original MILP and the two-stage MILP. Additionally, the results reveal that the solution quality deteriorates as the ratio of the set of mandatory visits increases or the maximum allowable travel time decreases. This study represents the first attempt to integrate mandatory visits into SOP, thereby establishing a new research direction in this area. The potential impact of this research is significant, as it introduces new possibilities for addressing complex combinatorial optimization problems. Full article

(This article belongs to the Section Engineering Mathematics)

16 pages, 2393 KiB

Open AccessArticle

A New Hybrid Descent Algorithm for Large-Scale Nonconvex Optimization and Application to Some Image Restoration Problems

by Shuai Wang, Xiaoliang Wang, Yuzhu Tian and Liping Pang

Mathematics 2024, 12(19), 3088; https://doi.org/10.3390/math12193088 - 2 Oct 2024

Abstract

Conjugate gradient methods are widely used and attractive for large-scale unconstrained smooth optimization problems, with simple computation, low memory requirements, and interesting theoretical information on the features of curvature. Based on the strongly convergent property of the Dai–Yuan method and attractive numerical performance [...] Read more.

Conjugate gradient methods are widely used and attractive for large-scale unconstrained smooth optimization problems, with simple computation, low memory requirements, and interesting theoretical information on the features of curvature. Based on the strongly convergent property of the Dai–Yuan method and attractive numerical performance of the Hestenes–Stiefel method, a new hybrid descent conjugate gradient method is proposed in this paper. The proposed method satisfies the sufficient descent property independent of the accuracy of the line search strategies. Under the standard conditions, the trust region property and the global convergence are established, respectively. Numerical results of 61 problems with 9 large-scale dimensions and 46 ill-conditioned matrix problems reveal that the proposed method is more effective, robust, and reliable than the other methods. Additionally, the hybrid method also demonstrates reliable results for some image restoration problems. Full article

(This article belongs to the Special Issue Optimization Algorithms: Theory and Applications)

18 pages, 522 KiB

Open AccessArticle

Uncovering Hidden Patterns: Approximate Resurgent Resummation from Truncated Series

by Alessio Maiezza and Juan Carlos Vasquez

Mathematics 2024, 12(19), 3087; https://doi.org/10.3390/math12193087 - 2 Oct 2024

Abstract

We analyze truncated series generated as divergent formal solutions of non-linear ordinary differential equations. Motivating the study is a specific non-linear, first-order differential equation, which is the basis of the resurgent formulation of renormalized perturbation theory in quantum field theory. We use the [...] Read more.

We analyze truncated series generated as divergent formal solutions of non-linear ordinary differential equations. Motivating the study is a specific non-linear, first-order differential equation, which is the basis of the resurgent formulation of renormalized perturbation theory in quantum field theory. We use the Borel–Padé approximant and classical analysis to determine the analytic structure of the solution using the first few terms of its asymptotic series. Afterward, we build an approximant, consistent with the resurgent properties of the equation. The procedure gives an approximate expression for the Borel–Ecalle resummation of the solution useful for practical applications. Connections with other physical applications are also discussed. Full article

(This article belongs to the Section Mathematical Physics)

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14 pages, 455 KiB

Open AccessArticle

On the Gradient Method in One Portfolio Management Problem

by Suriya Kumacheva and Vitalii Novgorodtcev

Mathematics 2024, 12(19), 3086; https://doi.org/10.3390/math12193086 - 2 Oct 2024

Abstract

This study refines the methodology for solving stochastic optimal control problems with quality criteria that include the sum of the quality functional of the classical formulation and an extremal measure. A two-level optimization solution of these kinds of problems is presented already for [...] Read more.

This study refines the methodology for solving stochastic optimal control problems with quality criteria that include the sum of the quality functional of the classical formulation and an extremal measure. A two-level optimization solution of these kinds of problems is presented already for the case where the quality functional consists only of the extremal measure. Our study shows the possibility of solving the original time inconsistency problem through solving a two-level optimization problem, where the outer problem is solved by gradient methods since the value function is convex and the inner problem is solved by classical methods. Some experiments were carried out and confirmed the validity of the theory. The results of the study can be applied to the case of portfolio management with quality criteria containing the Conditional Value-at-Risk (CVaR) metric. Full article

(This article belongs to the Special Issue Mathematical Modeling and Applications in Industrial Organization)

18 pages, 554 KiB

Open AccessArticle

Measure-Theoretic Analysis of Stochastic Competence Sets and Dynamic Shapley Values in Banach Spaces

by Jih-Jeng Huang and Chin-Yi Chen

Mathematics 2024, 12(19), 3085; https://doi.org/10.3390/math12193085 - 1 Oct 2024

Abstract

We develop a measure-theoretic framework for dynamic Shapley values in Banach spaces, extending classical cooperative game theory to continuous-time, infinite-dimensional settings. We prove the existence and uniqueness of strong solutions to stochastic differential equations modeling competence evolution in Banach spaces, establishing sample path [...] Read more.

We develop a measure-theoretic framework for dynamic Shapley values in Banach spaces, extending classical cooperative game theory to continuous-time, infinite-dimensional settings. We prove the existence and uniqueness of strong solutions to stochastic differential equations modeling competence evolution in Banach spaces, establishing sample path continuity and moment estimates. The dynamic Shapley value is rigorously defined as a càdlàg stochastic process with an axiomatic characterization. We derive a martingale representation for this process and establish its asymptotic properties, including a strong law of large numbers and a functional central limit theorem under α-mixing conditions. This framework provides a rigorous basis for analyzing dynamic value attribution in abstract spaces, with potential applications to economic and game-theoretic models. Full article

(This article belongs to the Special Issue Applied Mathematical Modelling and Dynamical Systems, 2nd Edition)

37 pages, 6091 KiB

Open AccessArticle

Abstract Cyclic Functional Relation and Taxonomies of Cyclic Signals Mathematical Models: Construction, Definitions and Properties

by Serhii Lupenko

Mathematics 2024, 12(19), 3084; https://doi.org/10.3390/math12193084 - 1 Oct 2024

Abstract

This work is devoted to the procedure of the construction of an abstract cyclic functional relation, which summarizes and extends the known results for a cyclically correlated random process and a cyclic (cyclically distributed) random process to the case of arbitrary cyclic functional [...] Read more.

This work is devoted to the procedure of the construction of an abstract cyclic functional relation, which summarizes and extends the known results for a cyclically correlated random process and a cyclic (cyclically distributed) random process to the case of arbitrary cyclic functional relations. Two alternative definitions of the abstract cyclic functional relation are given, and the fundamental properties of its cyclic and phase structures are presented. The theorem on the invariance of cyclicity attributes of an abstract cyclic functional relation to shifts of its argument, and which are determined by the rhythm function of this functional relation, is formulated and proved. This theorem gives the sufficient and necessary conditions that the rhythm function of an abstract cyclic functional relation must satisfy. By specifying the range of values and attributes of the cyclicity of an abstract cyclic functional relation, the definitions of important classes of cyclic functional relations are formulated. A deductive approach to building a wide system of taxonomies of classes of deterministic, stochastic, fuzzy and interval cyclic functional relations as potential mathematical models of cyclic signals is demonstrated. A comparative analysis of an abstract cyclic functional relation with the known mathematical models of cyclic signals was carried out. The results obtained in the article significantly expand and systematize the mathematical tools of the description of cyclic signals and are the basis for the development of effective model-based technologies for processing and computer simulation of signals with a cyclic space-time structure. Full article

(This article belongs to the Special Issue Advanced Information and Signal Processing: Models and Algorithms)

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