Mathematics

29 pages, 375 KB

Open AccessEditor’s ChoiceArticle

Spherical Harmonics and Gravity Intensity Modeling Related to a Special Class of Triaxial Ellipsoids

by Gerassimos Manoussakis and Panayiotis Vafeas

Mathematics 2025, 13(13), 2115; https://doi.org/10.3390/math13132115 - 27 Jun 2025

Viewed by 653

The G-modified Helmholtz equation is a partial differential equation that allows gravity intensity g to be expressed as a series of spherical harmonics, with the radial distance r raised to irrational powers. In this study, we consider a non-rotating triaxial ellipsoid parameterized by [...] Read more.

The G-modified Helmholtz equation is a partial differential equation that allows gravity intensity g to be expressed as a series of spherical harmonics, with the radial distance r raised to irrational powers. In this study, we consider a non-rotating triaxial ellipsoid parameterized by the geodetic latitude φ and geodetic longitude λ, and eccentricities e_e, e_x, e_y. On its surface, the value of gravity potential has a constant value, defining a level triaxial ellipsoid. In addition, the gravity intensity is known on the surface, which allows us to formulate a Dirichlet boundary value problem for determining the gravity intensity as a series of spherical harmonics. This expression for gravity intensity is presented here for the first time, filling a gap in the study of triaxial ellipsoids and spheroids. Given that the triaxial ellipsoid has very small eccentricities, a first order approximation can be made by retaining only the terms containing e_e² and e_x². The resulting expression in spherical harmonics contains even degree and even order harmonic coefficients, along with the associated Legendre functions. The maximum degree and order that occurs is four. Finally, as a special case, we present the geometrical degeneration of an oblate spheroid. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 2nd Edition)

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18 pages, 803 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Gaussian Process with Vine Copula-Based Context Modeling for Contextual Multi-Armed Bandits

by Jong-Min Kim

Mathematics 2025, 13(13), 2058; https://doi.org/10.3390/math13132058 - 21 Jun 2025

Cited by 1 | Viewed by 590

Abstract

We propose a novel contextual multi-armed bandit (CMAB) framework that integrates copula-based context generation with Gaussian Process (GP) regression for reward modeling, addressing complex dependency structures and uncertainty in sequential decision-making. Context vectors are generated using Gaussian and vine copulas to capture nonlinear [...] Read more.

We propose a novel contextual multi-armed bandit (CMAB) framework that integrates copula-based context generation with Gaussian Process (GP) regression for reward modeling, addressing complex dependency structures and uncertainty in sequential decision-making. Context vectors are generated using Gaussian and vine copulas to capture nonlinear dependencies, while arm-specific reward functions are modeled via GP regression with Beta-distributed targets. We evaluate three widely used bandit policies—Thompson Sampling (TS),

ε

-Greedy, and Upper Confidence Bound (UCB)—on simulated environments informed by real-world datasets, including Boston Housing and Wine Quality. The Boston Housing dataset exemplifies heterogeneous decision boundaries relevant to housing-related marketing, while the Wine Quality dataset introduces sensory feature-based arm differentiation. Our empirical results indicate that the

ε

-Greedy policy consistently achieves the highest cumulative reward and lowest regret across multiple runs, outperforming both GP-based TS and UCB in high-dimensional, copula-structured contexts. These findings suggest that combining copula theory with GP modeling provides a robust and flexible foundation for data-driven sequential experimentation in domains characterized by complex contextual dependencies. Full article

(This article belongs to the Special Issue Uncertainty Quantification Techniques in Statistics, Machine Learning and FinTech: 2nd Edition)

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12 pages, 1132 KB

Open AccessFeature PaperEditor’s ChoiceArticle

On the Relation Between Distances and Seminorms on Fréchet Spaces, with Application to Isometries

by Isabelle Chalendar, Lucas Oger and Jonathan R. Partington

Mathematics 2025, 13(13), 2053; https://doi.org/10.3390/math13132053 - 20 Jun 2025

Viewed by 302

Abstract

A study is made of linear isometries on Fréchet spaces for which the metric is given in terms of a sequence of seminorms. This establishes sufficient conditions on the growth of the function that defines the metric in terms of the seminorms to [...] Read more.

A study is made of linear isometries on Fréchet spaces for which the metric is given in terms of a sequence of seminorms. This establishes sufficient conditions on the growth of the function that defines the metric in terms of the seminorms to ensure that a linear operator preserving the metric also preserves each of these seminorms. As an application, characterizations are given of the isometries on various spaces including those of holomorphic functions on complex domains and continuous functions on open sets, extending the Banach–Stone theorem to surjective and nonsurjective cases. Full article

(This article belongs to the Section C4: Complex Analysis)

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22 pages, 1590 KB

Open AccessEditor’s ChoiceArticle

Continuous Exchangeable Markov Chains, Idempotent and 1-Dependent Copulas

by Martial Longla

Mathematics 2025, 13(12), 2034; https://doi.org/10.3390/math13122034 - 19 Jun 2025

Viewed by 1573

Abstract

New copula families are constructed based on orthogonality in

L^{2} (0, 1)

. Subclasses of idempotent copulas with square integrable densities are derived. It is shown that these copulas generate exchangeable Markov chains that behave as independent and identically [...] Read more.

New copula families are constructed based on orthogonality in

L^{2} (0, 1)

. Subclasses of idempotent copulas with square integrable densities are derived. It is shown that these copulas generate exchangeable Markov chains that behave as independent and identically distributed random variables conditionally on the initial variable. We prove that the extracted family of copulas is the only set of symmetric idempotent copulas with square integrable densities. We extend these copula families to asymmetric copulas with square integrable densities having special dependence properties. One of our extensions includes the Farlie–Gumbel–Morgenstern (FGM) copula family. The mixing properties of Markov chains generated by these copulas are established. The Spearman’s correlation coefficient

ρ_{S}

is provided for each of these copula families. Some graphs are also provided to illustrate the properties of the copula densities. Full article

(This article belongs to the Special Issue Advances in Statistical Modeling: Copulas, Large Sample Theory and Applications)

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25 pages, 338 KB

Open AccessEditor’s ChoiceArticle

Characterization of the Convergence Rate of the Augmented Lagrange for the Nonlinear Semidefinite Optimization Problem

by Yule Zhang, Jia Wu, Jihong Zhang and Haoyang Liu

Mathematics 2025, 13(12), 1946; https://doi.org/10.3390/math13121946 - 11 Jun 2025

Viewed by 523

Abstract

The convergence rate of the augmented Lagrangian method (ALM) for solving the nonlinear semidefinite optimization problem is studied. Under the Jacobian uniqueness conditions, when a multiplier vector

(π, Y)

and the penalty parameter

σ

are chosen such that

σ

is [...] Read more.

The convergence rate of the augmented Lagrangian method (ALM) for solving the nonlinear semidefinite optimization problem is studied. Under the Jacobian uniqueness conditions, when a multiplier vector

(π, Y)

and the penalty parameter

σ

are chosen such that

σ

is larger than a threshold

σ^{*} > 0

and the ratio

∥ (π, Y) - (π^{*}, Y^{*}) ∥ / σ

is small enough, it is demonstrated that the convergence rate of the augmented Lagrange method is linear with respect to

∥ (π, Y) - (π^{*}, Y^{*}) ∥

and the ratio constant is proportional to

1 / σ

, where

(π^{*}, Y^{*})

is the multiplier corresponding to a local minimizer. Furthermore, by analyzing the second-order derivative of the perturbation function of the nonlinear semidefinite optimization problem, we characterize the rate constant of local linear convergence of the sequence of Lagrange multiplier vectors produced by the augmented Lagrange method. This characterization shows that the sequence of Lagrange multiplier vectors has a Q-linear convergence rate when the sequence of penalty parameters

{σ_{k}}

has an upper bound and the convergence rate is superlinear when

{σ_{k}}

is increasing to infinity. Full article

(This article belongs to the Section D: Statistics and Operational Research)

28 pages, 1310 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Bridging Crisp-Set Qualitative Comparative Analysis and Association Rule Mining: A Formal and Computational Integration

by Acácio Dom Luís, Rafael Benítez and María del Carmen Bas

Mathematics 2025, 13(12), 1939; https://doi.org/10.3390/math13121939 - 11 Jun 2025

Viewed by 677

Abstract

In this paper, a novel mathematical formalization of Crisp-Set Qualitative Comparative Analysis (csQCA) that enables a rigorous connection with a specific class of association rule mining (ARM) problems is proposed. Although these two methodologies are frequently used to identify logical patterns in binary [...] Read more.

In this paper, a novel mathematical formalization of Crisp-Set Qualitative Comparative Analysis (csQCA) that enables a rigorous connection with a specific class of association rule mining (ARM) problems is proposed. Although these two methodologies are frequently used to identify logical patterns in binary datasets, they originate from different traditions. While csQCA is rooted in set theory and Boolean logic and is primarily applied in the social sciences to model causal complexity, ARM originates from data mining and is widely used to discover frequent co-occurrences among items. In this study, we establish a formal mathematical equivalence between csQCA configurations and a subclass of association rules, including both positive and negative conditions. Moreover, we propose a minimization procedure for association rules that mirrors the Quine–McCluskey reduction method employed in csQCA. We demonstrate the consistency of the results obtained using both methodologies through two examples (a small-N study on internet shutdowns in Sub-Saharan Africa and a large-N analysis of immigration attitudes in Europe) and some numerical experiments. However, it is also clear that ARM offers improved scalability and robustness in high-dimensional contexts. Overall, these findings provide researchers with valuable theoretical and practical guidance when choosing between these approaches in qualitative data analysis. Full article

(This article belongs to the Section E1: Mathematics and Computer Science)

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35 pages, 1074 KB

Open AccessEditor’s ChoiceArticle

oSets: Observer-Dependent Sets

by Mohamed Quafafou

Mathematics 2025, 13(12), 1928; https://doi.org/10.3390/math13121928 - 10 Jun 2025

Viewed by 548

Abstract

Sets play a foundational role in organizing, understanding, and interacting with the world in our daily lives. They also play a critical role in the functioning and behavior of social robots and artificial intelligence systems, which are designed to interact with humans and [...] Read more.

Sets play a foundational role in organizing, understanding, and interacting with the world in our daily lives. They also play a critical role in the functioning and behavior of social robots and artificial intelligence systems, which are designed to interact with humans and their environments in meaningful and socially intelligent ways. A multitude of non-classical set theories emerged during the last half-century aspiring to supplement Cantor’s set theory, allowing sets to be true to the reality of life by supporting, for example, human imprecision and uncertainty. The aim of this paper is to continue this effort of introducing oSets, which are sets depending on the perception of their observers. Our main objective is to align set theory with human cognition and perceptual diversity. In this context, an accessible set is a class of objects for which perception is passive, i.e., it is independent of perception; otherwise, it is called an oSet, which cannot be known exactly with respect to its observers, but it can only be approximated by a family of sets representing the diversity of its perception. Thus, the new introduced membership function is a three-place predicate denoted

\in_{i}

, where the expression “

x \in_{i} X

” indicates that the “observer” i perceives the element x as belonging to the set X. The accessibility notion is related to perception and can be best summarized as follows: “to be accessible is to be perceived”, presenting a weaker stance than Berkeley’s idealism, which asserts that “to be is to be perceived”. Full article

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44 pages, 12058 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Harmonizer: A Universal Signal Tokenization Framework for Multimodal Large Language Models

by Amin Amiri, Alireza Ghaffarnia, Nafiseh Ghaffar Nia, Dalei Wu and Yu Liang

Mathematics 2025, 13(11), 1819; https://doi.org/10.3390/math13111819 - 29 May 2025

Viewed by 2081

Abstract

This paper introduces Harmonizer, a universal framework designed for tokenizing heterogeneous input signals, including text, audio, and video, to enable seamless integration into multimodal large language models (LLMs). Harmonizer employs a unified approach to convert diverse, non-linguistic signals into discrete tokens via its [...] Read more.

This paper introduces Harmonizer, a universal framework designed for tokenizing heterogeneous input signals, including text, audio, and video, to enable seamless integration into multimodal large language models (LLMs). Harmonizer employs a unified approach to convert diverse, non-linguistic signals into discrete tokens via its FusionQuantizer architecture, built on FluxFormer, to efficiently capture essential signal features while minimizing complexity. We enhance features through STFT-based spectral decomposition, Hilbert transform analytic signal extraction, and SCLAHE spectrogram contrast optimization, and train using a composite loss function to produce reliable embeddings and construct a robust vector vocabulary. Experimental validation on music datasets such as E-GMD v1.0.0, Maestro v3.0.0, and GTZAN demonstrates high fidelity across 288 s of vocal signals (MSE = 0.0037, CC = 0.9282, Cosine Sim. = 0.9278, DTW = 12.12, MFCC Sim. = 0.9997, Spectral Conv. = 0.2485). Preliminary tests on text reconstruction and UCF-101 video clips further confirm Harmonizer’s applicability across discrete and spatiotemporal modalities. Rooted in the universality of wave phenomena and Fourier theory, Harmonizer offers a physics-inspired, modality-agnostic fusion mechanism via wave superposition and interference principles. In summary, Harmonizer integrates natural language processing and signal processing into a coherent tokenization paradigm for efficient, interpretable multimodal learning. Full article

(This article belongs to the Special Issue Applied Mathematics in Machine Learning and Cloud Computing: Foundations and Applications)

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23 pages, 1106 KB

Open AccessFeature PaperEditor’s ChoiceArticle

A Quantum Algorithm for the Classification of Patterns of Boolean Functions

by Theodore Andronikos, Constantinos Bitsakos, Konstantinos Nikas, Georgios I. Goumas and Nectarios Koziris

Mathematics 2025, 13(11), 1750; https://doi.org/10.3390/math13111750 - 25 May 2025

Cited by 1 | Viewed by 704

Abstract

This paper introduces a novel quantum algorithm that is able to classify a hierarchy of classes of imbalanced Boolean functions. The fundamental characteristic of imbalanced Boolean functions is that the proportion of elements in their domain that take the value 0 is not [...] Read more.

This paper introduces a novel quantum algorithm that is able to classify a hierarchy of classes of imbalanced Boolean functions. The fundamental characteristic of imbalanced Boolean functions is that the proportion of elements in their domain that take the value 0 is not equal to the proportion of elements that take the value 1. For every positive integer, n, the hierarchy contains a class of n-ary Boolean functions defined according to their behavioral pattern. The common trait of all the functions belonging to the same class is that they possess the same imbalance ratio. Our algorithm achieves classification in a straightforward manner as the final measurement reveals the unknown function with a probability of

1.0

. Let us also note that the proposed algorithm is an optimal oracular algorithm because it can classify the aforementioned functions with just a single query to the oracle. At the same time, we explain in detail the methodology we followed to design this algorithm in the hope that it will prove general and fruitful, given that it can be easily modified and extended to address other classes of imbalanced Boolean functions that exhibit different behavioral patterns. Full article

(This article belongs to the Special Issue Quantum Computing and Networking)

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23 pages, 2863 KB

Open AccessEditor’s ChoiceArticle

Using Physics-Informed Neural Networks for Modeling Biological and Epidemiological Dynamical Systems

by Amer Farea, Olli Yli-Harja and Frank Emmert-Streib

Mathematics 2025, 13(10), 1664; https://doi.org/10.3390/math13101664 - 19 May 2025

Cited by 1 | Viewed by 3741

Abstract

Physics-Informed Neural Networks (PINNs) have emerged as a powerful approach for integrating physical laws into a deep learning framework, offering enhanced capabilities for solving complex problems. Despite their potential, the practical implementation of PINNs presents significant challenges. This paper explores the application of [...] Read more.

Physics-Informed Neural Networks (PINNs) have emerged as a powerful approach for integrating physical laws into a deep learning framework, offering enhanced capabilities for solving complex problems. Despite their potential, the practical implementation of PINNs presents significant challenges. This paper explores the application of PINNs to systems of ordinary differential equations (ODEs), focusing on two key challenges: the forward problem of solution approximation and the inverse problem of parameter estimation. We present three detailed case studies involving dynamical systems for tumor growth, gene expression, and the SIR (Susceptible, Infected, Recovered) model for disease spread. This paper outlines the core principles of PINNs and their integration with physical laws during neural network training. It details the steps involved in implementing PINNs, emphasizing the critical role of network architecture and hyperparameter tuning in achieving optimal performance. Additionally, we provide a Python package, ODE-PINN, to reproduce results for ODE-based systems. Our findings demonstrate that PINNs can yield accurate and consistent solutions, but their performance is highly sensitive to network architecture and hyperparameters tuning. These results underscore the need for customized configurations and robust optimization strategies. Overall, this study confirms the significant potential of PINNs to advance the understanding of dynamical systems in biology and epidemiology. Full article

(This article belongs to the Special Issue Machine Learning Methods and Mathematical Modeling with Applications)

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20 pages, 3731 KB

Open AccessEditor’s ChoiceArticle

Fuzzy Clustering with Uninorm-Based Distance Measure

by Evgeny Kagan, Alexander Novoselsky and Alexander Rybalov

Mathematics 2025, 13(10), 1661; https://doi.org/10.3390/math13101661 - 19 May 2025

Viewed by 768

Abstract

In this paper, we suggest an algorithm of fuzzy clustering with a uninorm-based distance measure. The algorithm follows a general scheme of fuzzy c-means (FCM) clustering, but in contrast to the existing algorithm, it implements logical distance between data instances. The centers [...] Read more.

In this paper, we suggest an algorithm of fuzzy clustering with a uninorm-based distance measure. The algorithm follows a general scheme of fuzzy c-means (FCM) clustering, but in contrast to the existing algorithm, it implements logical distance between data instances. The centers of the clusters calculated by the algorithm are less dispersed and are concentrated in the areas of the actual centers of the clusters that result in the more accurate recognition of the number of clusters and of data structure. Full article

(This article belongs to the Special Issue Advances in Multi-Criteria Decision Making Methods with Applications)

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29 pages, 2898 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Distributed Observer-Based Adaptive Trajectory Tracking and Formation Control for the Swarm of Nonholonomic Mobile Robots with Unknown Wheel Slippage

by Sathishkumar Moorthy, Sachin Sakthi Kuppusami Sakthivel, Young Hoon Joo and Jae Hoon Jeong

Mathematics 2025, 13(10), 1628; https://doi.org/10.3390/math13101628 - 15 May 2025

Cited by 2 | Viewed by 662

Abstract

Nonholonomoic mobile robots (NMRs) are widely used in logistics transportation and industrial production, with motion control remaining a key focus in current WMR research. However, most previously developed controllers assume ideal conditions without considering motion slippage. Neglecting slippage factors often leads to reduced [...] Read more.

Nonholonomoic mobile robots (NMRs) are widely used in logistics transportation and industrial production, with motion control remaining a key focus in current WMR research. However, most previously developed controllers assume ideal conditions without considering motion slippage. Neglecting slippage factors often leads to reduced control performance, causing instability and deviation from the robot’s path. To address such a challenge, this paper proposes an intelligent method for estimating the longitudinal wheel slip, enabling effective compensation for the adverse effects of slippage. The proposed algorithm relies on the development of an adaptive trajectory tracking controller for the leader robot. This controller enables the leader robot to accurately follow a virtual reference trajectory while estimating the actual slipping ratio with precision. By employing this approach, the mobile robot can effectively address the challenge of wheel slipping and enhance its overall performance. Next, a distributed observer is developed for each NMR that uses both its own and adjacent robot’s information to determine the leader’s state. To solve this difficulty for the follower robot to receive the states of the leader in a large group of robots, distributed formation controllers are designed. Further, Lyapunov stability theory is utilized to analyze the convergence of tracking errors that guarantees multi-robot formation. At last, numerical simulations on a group of NMR are provided to illustrate the performance of the designed controller. The leader robot achieved a low RMSE of 1.7571, indicating accurate trajectory tracking. Follower robots showed RMSEs of 2.7405 (Robot 2), 3.0789 (Robot 4), and 4.3065 (Robot 3), reflecting minor variations due to the distributed control strategy and local disturbances. Full article

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20 pages, 1271 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Advanced Mathematical Modeling of Hydrogen and Methane Production in a Two-Stage Anaerobic Co-Digestion System

by Olympia Roeva, Elena Chorukova and Lyudmila Kabaivanova

Mathematics 2025, 13(10), 1601; https://doi.org/10.3390/math13101601 - 13 May 2025

Cited by 1 | Viewed by 529

Abstract

This study introduces a novel mathematical model characterizing the anaerobic co-digestion of wheat straw and waste algal biomass for hydrogen and methane production, implemented in a two-stage bioreactor system. Co-digestion can be a tool to increase biogas production utilizing difficult-to-digest organic waste by [...] Read more.

This study introduces a novel mathematical model characterizing the anaerobic co-digestion of wheat straw and waste algal biomass for hydrogen and methane production, implemented in a two-stage bioreactor system. Co-digestion can be a tool to increase biogas production utilizing difficult-to-digest organic waste by introducing easily degradable substrates. Two continuous operational regimes, with organic loading rates of 50 g/L and 33 g/L, were employed to generate the experimental datasets for model parameterization and validation, respectively. Parameter identification was achieved through dynamic experimentation, utilizing three distinct optimization algorithms: the deterministic active-set method (A-S) and the metaheuristics–genetic algorithm (GA), coyote optimization algorithm (COA), and marine predator algorithm (MPA). We assessed the predictive capability of the developed mathematical models using an independent dataset. The models demonstrated good agreement with the experimental data across all measured process variables. Notably, the MPA exhibited superior data fitting accuracy, as quantitatively confirmed by the objective function value, compared to GA, COA, and the A-S algorithm. Full article

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11 pages, 220 KB

Open AccessFeature PaperEditor’s ChoiceArticle

A Multi-Period Optimization Framework for Portfolio Selection Using Interval Analysis

by Florentin Șerban

Mathematics 2025, 13(10), 1552; https://doi.org/10.3390/math13101552 - 8 May 2025

Cited by 1 | Viewed by 741

Abstract

This paper presents a robust multi-period portfolio optimization framework that integrates interval analysis, entropy-based diversification, and downside risk control. In contrast to classical models relying on precise probabilistic assumptions, our approach captures uncertainty through interval-valued parameters for asset returns, risk, and liquidity—particularly suitable [...] Read more.

This paper presents a robust multi-period portfolio optimization framework that integrates interval analysis, entropy-based diversification, and downside risk control. In contrast to classical models relying on precise probabilistic assumptions, our approach captures uncertainty through interval-valued parameters for asset returns, risk, and liquidity—particularly suitable for volatile markets such as cryptocurrencies. The model seeks to maximize terminal portfolio wealth over a finite investment horizon while ensuring compliance with return, risk, liquidity, and diversification constraints at each rebalancing stage. Risk is modeled using semi-absolute deviation, which better reflects investor sensitivity to downside outcomes than variance-based measures, and diversification is promoted through Shannon entropy to prevent excessive concentration. A nonlinear multi-objective formulation ensures computational tractability while preserving decision realism. To illustrate the practical applicability of the proposed framework, a simulated case study is conducted on four major cryptocurrencies—Bitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB). The model evaluates three strategic profiles based on investor risk attitude: pessimistic (lower return bounds and upper risk bounds), optimistic (upper return bounds and lower risk bounds), and mixed (average values). The resulting final terminal wealth intervals are [1085.32, 1163.77] for the pessimistic strategy, [1123.89, 1245.16] for the mixed strategy, and [1167.42, 1323.55] for the optimistic strategy. These results demonstrate the model’s adaptability to different investor preferences and its empirical relevance in managing uncertainty under real-world volatility conditions. Full article

(This article belongs to the Section E: Applied Mathematics)

19 pages, 5504 KB

Open AccessEditor’s ChoiceArticle

Progressive Domain Decomposition for Efficient Training of Physics-Informed Neural Network

by Dawei Luo, Soo-Ho Jo and Taejin Kim

Mathematics 2025, 13(9), 1515; https://doi.org/10.3390/math13091515 - 4 May 2025

Cited by 2 | Viewed by 1799

Abstract

This study proposes a strategy for decomposing the computational domain to solve differential equations using physics-informed neural networks (PINNs) and progressively saving the trained model in each subdomain. The proposed progressive domain decomposition (PDD) method segments the domain based on the dynamics of [...] Read more.

This study proposes a strategy for decomposing the computational domain to solve differential equations using physics-informed neural networks (PINNs) and progressively saving the trained model in each subdomain. The proposed progressive domain decomposition (PDD) method segments the domain based on the dynamics of residual loss, thereby indicating the complexity of different sections within the entire domain. By analyzing residual loss pointwise and aggregating it over specific intervals, we identify critical regions requiring focused attention. This strategic segmentation allows for the application of tailored neural networks in identified subdomains, each characterized by varying levels of complexity. Additionally, the proposed method trains and saves the model progressively based on performance metrics, thereby conserving computational resources in sections where satisfactory results are achieved during the training process. The effectiveness of PDD is demonstrated through its application to complex PDEs, where it significantly enhances accuracy and conserves computational power by strategically simplifying the computational tasks into manageable segments. Full article

(This article belongs to the Special Issue Advanced Modeling and Design of Vibration and Wave Systems)

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39 pages, 7701 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Macroelement Analysis in T-Patches Using Lagrange Polynomials

by Christopher Provatidis and Sascha Eisenträger

Mathematics 2025, 13(9), 1498; https://doi.org/10.3390/math13091498 - 30 Apr 2025

Cited by 2 | Viewed by 721

Abstract

This paper investigates the derivation of global shape functions in T-meshed quadrilateral patches through transfinite interpolation and local elimination. The same shape functions may be alternatively derived starting from a background tensor product of Lagrange polynomials and then imposing linear constraints. Based on [...] Read more.

This paper investigates the derivation of global shape functions in T-meshed quadrilateral patches through transfinite interpolation and local elimination. The same shape functions may be alternatively derived starting from a background tensor product of Lagrange polynomials and then imposing linear constraints. Based on the nodal points of the T-mesh, which are associated with the primary degrees of freedom (DOFs), all the other points of the background grid (i.e., the secondary DOFs) are interpolated along horizontal and vertical stations (isolines) of the tensor product, and thus, linear relationships are derived. By implementing these constraints into the original formula/expression, global shape functions, which are only associated with primary DOFs, are created. The quality of the elements is verified by the numerical solution of a typical potential problem of second order, with boundary conditions of Dirichlet and Neumann type. Full article

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20 pages, 330 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Line Defects in One-Dimensional Hexagonal Quasicrystals

by Markus Lazar

Mathematics 2025, 13(9), 1493; https://doi.org/10.3390/math13091493 - 30 Apr 2025

Cited by 1 | Viewed by 591

Abstract

Using the eight-dimensional framework of the integral formalism of one-dimensional quasicrystals, the analytical expressions for the displacement fields and stress functions of line defects, which are dislocations and line forces, in one-dimensional hexagonal quasicrystals of Laue class 10 are derived. The self-energy of [...] Read more.

Using the eight-dimensional framework of the integral formalism of one-dimensional quasicrystals, the analytical expressions for the displacement fields and stress functions of line defects, which are dislocations and line forces, in one-dimensional hexagonal quasicrystals of Laue class 10 are derived. The self-energy of a straight dislocation, the self-energy of a line force, the Peach–Koehler force between two straight dislocations, and the Cherepanov force between two straight line forces in one-dimensional hexagonal quasicrystals of Laue class 10 are calculated. In addition, the two-dimensional Green tensor of one-dimensional hexagonal quasicrystals of Laue class 10 is given within the framework of the integral formalism. Full article

(This article belongs to the Special Issue Multiscale Mathematical Modeling)

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21 pages, 428 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Accelerated Numerical Simulations of a Reaction-Diffusion- Advection Model Using Julia-CUDA

by Angelo Ciaramella, Davide De Angelis, Pasquale De Luca and Livia Marcellino

Mathematics 2025, 13(9), 1488; https://doi.org/10.3390/math13091488 - 30 Apr 2025

Cited by 1 | Viewed by 603

Abstract

The emergence of exascale computing systems presents both opportunities and challenges in scientific computing, particularly for complex mathematical models requiring high-performance implementations. This paper addresses these challenges in the context of biomedical applications, specifically focusing on tumor angiogenesis modeling. We present a parallel [...] Read more.

The emergence of exascale computing systems presents both opportunities and challenges in scientific computing, particularly for complex mathematical models requiring high-performance implementations. This paper addresses these challenges in the context of biomedical applications, specifically focusing on tumor angiogenesis modeling. We present a parallel implementation for solving a system of partial differential equations that describe the dynamics of tumor-induced blood vessel formation. Our approach leverages the Julia programming language and its CUDA capabilities, combining a high-level paradigm with efficient GPU acceleration. The implementation incorporates advanced optimization strategies for memory management and kernel organization, demonstrating significant performance improvements for large-scale simulations while maintaining numerical accuracy. Experimental results confirm the performance gains and reliability of the proposed parallel implementation. Full article

(This article belongs to the Special Issue Advances in Numerical Mathematics for High-Performance Computing in the Exascale Era)

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20 pages, 2586 KB

Open AccessEditor’s ChoiceArticle

An In-Depth Investigation of the Riemann Zeta Function Using Infinite Numbers

by Emmanuel Thalassinakis

Mathematics 2025, 13(9), 1483; https://doi.org/10.3390/math13091483 - 30 Apr 2025

Viewed by 3482

Abstract

This study focuses on an in-depth investigation of the Riemann zeta function. For this purpose, infinite numbers and rotational infinite numbers, which have been introduced in previous studies published by the author, are used. These numbers are a powerful tool for solving problems [...] Read more.

This study focuses on an in-depth investigation of the Riemann zeta function. For this purpose, infinite numbers and rotational infinite numbers, which have been introduced in previous studies published by the author, are used. These numbers are a powerful tool for solving problems involving infinity that are otherwise difficult to solve. Infinite numbers are a superset of complex numbers and can be either complex numbers or some quantification of infinity. The Riemann zeta function can be written as a sum of three rotational infinite numbers, each of which represents infinity. Using these infinite numbers and their properties, a correlation of the non-trivial zeros of the Riemann zeta function with each other is revealed and proven. In addition, an interesting relation between the Euler–Mascheroni constant (γ) and the non-trivial zeros of the Riemann zeta function is proven. Based on this analysis, complex series limits are calculated and important conclusions about the Riemann zeta function are drawn. It turns out that when we have non-trivial zeros of the Riemann zeta function, the corresponding Dirichlet series increases linearly, in contrast to the other cases where this series also includes a fluctuating term. The above theoretical results are fully verified using numerical computations. Furthermore, a new numerical method is presented for calculating the non-trivial zeros of the Riemann zeta function, which lie on the critical line. In summary, by using infinite numbers, aspects of the Riemann zeta function are explored and revealed from a different perspective; additionally, interesting mathematical relationships that are difficult or impossible to solve with other methods are easily analyzed and solved. Full article

(This article belongs to the Special Issue Special Functions with Applications)

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18 pages, 338 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Fundamental Matrix, Measure Resolvent Kernel and Stability Properties of Fractional Linear Delayed System with Discontinuous Initial Conditions

by Hristo Kiskinov, Mariyan Milev, Milena Petkova and Andrey Zahariev

Mathematics 2025, 13(9), 1408; https://doi.org/10.3390/math13091408 - 25 Apr 2025

Viewed by 431

Abstract

In the present work, a Cauchy (initial) problem for a fractional linear system with distributed delays and Caputo-type derivatives of incommensurate order is considered. As the main result, a new straightforward approach to study the considered initial problem via an equivalent Volterra–Stieltjes integral [...] Read more.

In the present work, a Cauchy (initial) problem for a fractional linear system with distributed delays and Caputo-type derivatives of incommensurate order is considered. As the main result, a new straightforward approach to study the considered initial problem via an equivalent Volterra–Stieltjes integral system is introduced. This approach is based on the existence and uniqueness of a global fundamental matrix for the corresponding homogeneous system, which allows us to prove that the corresponding resolvent system possesses a unique measure resolvent kernel. As a consequence, an integral representation of the solutions of the studied system is obtained. Then, using the obtained results, relations between the stability of the zero solution of the homogeneous system and different kinds of boundedness of its other solutions are established. Full article

(This article belongs to the Section C: Mathematical Analysis)

33 pages, 458 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Recent Developments in Chen’s Biharmonic Conjecture and Some Related Topics

by Bang-Yen Chen

Mathematics 2025, 13(9), 1417; https://doi.org/10.3390/math13091417 - 25 Apr 2025

Viewed by 1175

Abstract

The study of biharmonic submanifolds in Euclidean spaces was introduced in the middle of the 1980s by the author in his program studying finite-type submanifolds. He defined biharmonic submanifolds in Euclidean spaces as submanifolds whose position vector field (x) satisfies the [...] Read more.

The study of biharmonic submanifolds in Euclidean spaces was introduced in the middle of the 1980s by the author in his program studying finite-type submanifolds. He defined biharmonic submanifolds in Euclidean spaces as submanifolds whose position vector field (x) satisfies the biharmonic equation, i.e.,

Δ^{2} x = 0

. A well-known conjecture proposed by the author in 1991 on biharmonic submanifolds states that every biharmonic submanifold of a Euclidean space is minimal, well known today as Chen’s biharmonic conjecture. On the other hand, independently, G.-Y. Jiang investigated biharmonic maps between Riemannian manifolds as the critical points of the bi-energy functional. In 2002, R. Caddeo, S. Montaldo, and C. Oniciuc pointed out that both definitions of biharmonicity of the author and G.-Y. Jiang coincide for the class of Euclidean submanifolds. Since then, the study of biharmonic submanifolds and biharmonic maps has attracted many researchers, and many interesting results have been achieved. A comprehensive survey of important results on this conjecture and on many related topics was presented by Y.-L. Ou and B.-Y. Chen in their 2020 book. The main purpose of this paper is to provide a detailed survey of recent developments in those subjects after the publication of Ou and Chen’s book. Full article

(This article belongs to the Special Issue Analysis on Differentiable Manifolds)

11 pages, 239 KB

Open AccessEditor’s ChoiceArticle

Resolving an Open Problem on the Exponential Arithmetic–Geometric Index of Unicyclic Graphs

by Kinkar Chandra Das and Jayanta Bera

Mathematics 2025, 13(9), 1391; https://doi.org/10.3390/math13091391 - 24 Apr 2025

Cited by 1 | Viewed by 404

Abstract

Recently, the exponential arithmetic–geometric index (

E A G

) was introduced. The exponential arithmetic–geometric index (

E A G

) of a graph G is defined as [...] Read more.

Recently, the exponential arithmetic–geometric index (

E A G

) was introduced. The exponential arithmetic–geometric index (

E A G

) of a graph G is defined as

E A G (G) = \sum_{v_{i} v_{j} \in E (G)} e^{\frac{d_{i} + d_{j}}{2 \sqrt{d_{i} d_{j}}}}

, where

d_{i}

represents the degree of the vertex

v_{i}

in G. The characterization of extreme structures in relation to graph invariants from the class of unicyclic graphs is an important problem in discrete mathematics. Cruz et al., 2022 proposed a unified method for finding extremal unicyclic graphs for exponential degree-based graph invariants. However, in the case of

E A G

, this method is insufficient for generating the maximal unicyclic graph. Consequently, the same article presented an open problem for the investigation of the maximal unicyclic graph with respect to this invariant. This article completely characterizes the maximal unicyclic graph in relation to

E A G

. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 2nd Edition)

19 pages, 1414 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Wavelet and Deep Learning Framework for Predicting Commodity Prices Under Economic and Financial Uncertainty

by Lyubov Doroshenko, Loretta Mastroeni and Alessandro Mazzoccoli

Mathematics 2025, 13(8), 1346; https://doi.org/10.3390/math13081346 - 20 Apr 2025

Cited by 3 | Viewed by 1500

Abstract

The analysis of commodity markets—particularly in the energy and metals sectors—is essential for understanding economic dynamics and guiding decision-making. Financial and economic uncertainty indices provide valuable insights that help reduce price uncertainty. This study employs wavelet analyses and wavelet energy-based measures to investigate [...] Read more.

The analysis of commodity markets—particularly in the energy and metals sectors—is essential for understanding economic dynamics and guiding decision-making. Financial and economic uncertainty indices provide valuable insights that help reduce price uncertainty. This study employs wavelet analyses and wavelet energy-based measures to investigate the relationship between these indices and commodity prices across multiple time scales. The wavelet approach captures complex, time-varying dependencies, offering a more nuanced understanding of how uncertainty indices influence commodity price fluctuations. By integrating this analysis with predictability measures, we assess how uncertainty indices enhance forecasting accuracy. We further incorporate deep learning models capable of capturing sequential patterns in financial time series into our analysis to better evaluate their predictive potential. Our findings highlight the varying impact of financial and economic uncertainty on the predictability of commodity prices, showing that while some indices offer valuable forecasting information, others display strong correlations without significant predictive power. These results underscore the need for tailored predictive models, as different commodities react differently to the same financial conditions. By combining wavelet-based measures with machine learning techniques, this study presents a comprehensive framework for evaluating the role of uncertainty in commodity markets. The insights gained can support investors, policymakers, and market analysts in making more informed decisions. Full article

(This article belongs to the Special Issue Advancements in Applied Mathematics for Economic Data Analytics: Models, Methods, and Applications)

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17 pages, 457 KB

Open AccessEditor’s ChoiceArticle

Hyperbolic Representation of the Richards Growth Model

by Marcin Molski

Mathematics 2025, 13(8), 1316; https://doi.org/10.3390/math13081316 - 17 Apr 2025

Viewed by 634

Abstract

The phenomenological universalities (PU) approach is employed to derive the Richards growth function in the unknown hyperbolic representation. The formula derived can be applied in theoretical modeling of sigmoid and involuted growth of biological systems. In the model proposed, the exponent in the [...] Read more.

The phenomenological universalities (PU) approach is employed to derive the Richards growth function in the unknown hyperbolic representation. The formula derived can be applied in theoretical modeling of sigmoid and involuted growth of biological systems. In the model proposed, the exponent in the Richards function has the following clear biological meaning: it describes the number of cells doubling, leading to an increase in a biomass of the system from

m_{0}

(birth or hatching mass) to the limiting value

m_{\infty}

(mass at maturity). The generalized form of the universal growth function is derived. It can be employed in fitting the weight–age data for a variety of biological systems, including copepods, tumors, fish, birds, mammals and dinosaurs. Both the PU methodology and the Richards model can be effectively applied in the theoretical modeling of infectious disease outbreaks. To substantiate this assertion, the simplest PU-SIR (Susceptible–Infective–Removed) epidemiological model is considered. In this approach, it is assumed that the number of births is approximately equal to the number of deaths, while the impact of recovered (quarantined) individuals on the dynamics of the infection is negligible. Full article

(This article belongs to the Section E3: Mathematical Biology)

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14 pages, 360 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Prime Strictly Concentric Magic Squares of Odd Order

by Anna Louise Skelt, Stephanie Perkins and Paul Alun Roach

Mathematics 2025, 13(8), 1261; https://doi.org/10.3390/math13081261 - 11 Apr 2025

Cited by 1 | Viewed by 775

Abstract

Magic squares have been widely studied, with publications of mathematical interest dating back over 100 years. Most studies construct and analyse specific subsets of magic squares, with some exploring links to puzzles, number theory, and graph theory. The subset of magic squares this [...] Read more.

Magic squares have been widely studied, with publications of mathematical interest dating back over 100 years. Most studies construct and analyse specific subsets of magic squares, with some exploring links to puzzles, number theory, and graph theory. The subset of magic squares this paper focuses on are those termed prime strictly concentric magic squares (PSCMS), and their general definitions, examples, and important properties are also presented. Previously, only the minimum centre cell values of PSCMS of odd order 5 to 19 were presented, by Makarova in 2015. In this paper, the corresponding list of primes for all minimum PSCMS of order 5 is given, and the number of minimum PSCMS of order 5 is enumerated. Full article

(This article belongs to the Section A: Algebra and Logic)

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14 pages, 515 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Set-Valued Approximation—Revisited and Improved

by David Levin

Mathematics 2025, 13(7), 1194; https://doi.org/10.3390/math13071194 - 4 Apr 2025

Viewed by 434

Abstract

We address the problem of approximating a set-valued function F, where

F : [a, b] \to K (R^{d})

given its samples

{F (a + i h)}_{i = 0}^{N}

, with [...] Read more.

We address the problem of approximating a set-valued function F, where

F : [a, b] \to K (R^{d})

given its samples

{F (a + i h)}_{i = 0}^{N}

, with

h = (b - a) / N

. We revisit an existing method that approximates set-valued functions by interpolating signed-distance functions. This method provides a high-order approximation for general topologies but loses accuracy near points where F undergoes topological changes. To address this, we introduce new techniques that enhance efficiency and maintain high-order accuracy across

[a, b]

. Building on the foundation of previous publication, we introduce new techniques to improve the method’s efficiency and extend its high-order approximation accuracy throughout the entire interval

[a, b]

. Particular focus is placed on identifying and analyzing the behavior of F near topological transition points. To address this, two algorithms are introduced. The first algorithm employs signed-distance quasi-interpolation, incorporating specialized adjustments to effectively handle singularities at points of topological change. The second algorithm leverages an implicit function representation of

G r a p h (F)

, offering an alternative and robust approach to its approximation. These enhancements improve accuracy and stability in handling set-valued functions with changing topologies. Full article

(This article belongs to the Special Issue Advances in Approximation Theory and Numerical Functional Analysis)

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24 pages, 2090 KB

Open AccessFeature PaperEditor’s ChoiceReview

On Regime Switching Models

by Zhenni Tan and Yuehua Wu

Mathematics 2025, 13(7), 1128; https://doi.org/10.3390/math13071128 - 29 Mar 2025

Cited by 4 | Viewed by 5109

Abstract

Regime switching models have been widely studied for their ability to capture the dynamic behavior of time series data and are widely used in economic and financial data analysis. This paper reviews various regime switching models with various regime switching mechanisms, including threshold [...] Read more.

Regime switching models have been widely studied for their ability to capture the dynamic behavior of time series data and are widely used in economic and financial data analysis. This paper reviews various regime switching models with various regime switching mechanisms, including threshold models, hidden Markov regime switching models, hidden semi-Markov regime switching models, and smooth transition models. The focus is on regime switching models for time series, studying their underlying frameworks, popular variants, and commonly used estimation methods. In addition, six different regime switching models are compared using two real-world datasets. Full article

(This article belongs to the Special Issue Advanced Statistical Applications in Financial Econometrics)

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25 pages, 2066 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Is π a Chaos Generator?

by Natalia Petrovskaya

Mathematics 2025, 13(7), 1126; https://doi.org/10.3390/math13071126 - 29 Mar 2025

Viewed by 588

Abstract

We consider a circular motion problem related to blind search in confined space. A particle moves in a unit circle in discrete time to find the escape channel and leave the circle through it. We first explain how the exit time depends on [...] Read more.

We consider a circular motion problem related to blind search in confined space. A particle moves in a unit circle in discrete time to find the escape channel and leave the circle through it. We first explain how the exit time depends on the initial position of the particle when the channel width is fixed. We then investigate how narrowing the channel moves the system from discrete changes in the exit time to the ultimate ‘countable chaos’ state that arises in the problem when the channel width becomes infinitely small. It will be shown in the paper that inherent randomness exists in the problem due to the nature of circular motion as the number

π

acts as a random number generator in the system. Randomness of the decimal digits of

π

results in sensitive dependence on initial conditions in the system with an infinitely narrow channel, and we argue that even a simple linear dynamical system can exhibit features of chaotic behaviour, provided that the system has inherent noise. Full article

(This article belongs to the Special Issue Applied Mathematics in Nonlinear Dynamics and Chaos)

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17 pages, 310 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Conditional Quantization for Uniform Distributions on Line Segments and Regular Polygons

by Pigar Biteng, Mathieu Caguiat, Tsianna Dominguez and Mrinal Kanti Roychowdhury

Mathematics 2025, 13(7), 1024; https://doi.org/10.3390/math13071024 - 21 Mar 2025

Cited by 3 | Viewed by 409

Abstract

Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. If, in the quantization some of the elements in the support are preselected, then the quantization [...] Read more.

Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. If, in the quantization some of the elements in the support are preselected, then the quantization is called a conditional quantization. In this paper, we investigate the conditional quantization for the uniform distributions defined on the unit line segments and m-sided regular polygons, where

m \geq 3

, inscribed in a unit circle. Full article

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14 pages, 495 KB

Open AccessEditor’s ChoiceArticle

A Fast Projected Gradient Algorithm for Quaternion Hermitian Eigenvalue Problems

by Shan-Qi Duan and Qing-Wen Wang

Mathematics 2025, 13(6), 994; https://doi.org/10.3390/math13060994 - 18 Mar 2025

Cited by 3 | Viewed by 694

Abstract

In this paper, based on the novel generalized Hamilton-real (GHR) calculus, we propose for the first time a quaternion Nesterov accelerated projected gradient algorithm for computing the dominant eigenvalue and eigenvector of quaternion Hermitian matrices. By introducing momentum terms and look-ahead updates, the [...] Read more.

In this paper, based on the novel generalized Hamilton-real (GHR) calculus, we propose for the first time a quaternion Nesterov accelerated projected gradient algorithm for computing the dominant eigenvalue and eigenvector of quaternion Hermitian matrices. By introducing momentum terms and look-ahead updates, the algorithm achieves a faster convergence rate. We theoretically prove the convergence of the quaternion Nesterov accelerated projected gradient algorithm. Numerical experiments show that the proposed method outperforms the quaternion projected gradient ascent method and the traditional algebraic methods in terms of computational accuracy and runtime efficiency. Full article

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21 pages, 1033 KB

Open AccessFeature PaperEditor’s ChoiceArticle

On Strongly Regular Graphs and the Friendship Theorem

by Igal Sason

Mathematics 2025, 13(6), 970; https://doi.org/10.3390/math13060970 - 14 Mar 2025

Cited by 1 | Viewed by 2546

Abstract

This paper presents an alternative proof of the celebrated friendship theorem, originally established by Erdős, Rényi, and Sós in 1966. The proof relies on a closed-form expression for the Lovász

ϑ

-function of strongly regular graphs, recently derived by the author. Additionally, this [...] Read more.

This paper presents an alternative proof of the celebrated friendship theorem, originally established by Erdős, Rényi, and Sós in 1966. The proof relies on a closed-form expression for the Lovász

ϑ

-function of strongly regular graphs, recently derived by the author. Additionally, this paper considers some known extensions of the theorem, offering discussions that provide insights into the friendship theorem, one of its extensions, and the proposed proof. Leveraging the closed-form expression for the Lovász

ϑ

-function of strongly regular graphs, this paper further establishes new necessary conditions for a strongly regular graph to be a spanning or induced subgraph of another strongly regular graph. In the case of induced subgraphs, the analysis also incorporates a property of graph energies. Some of these results are extended to regular graphs and their subgraphs. Full article

(This article belongs to the Special Issue Advances in Combinatorics, Discrete Mathematics and Graph Theory)

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16 pages, 248 KB

Open AccessEditor’s ChoiceArticle

Conformal Interactions of Osculating Curves on Regular Surfaces in Euclidean 3-Space

by Yingxin Cheng, Yanlin Li, Pushpinder Badyal, Kuljeet Singh and Sandeep Sharma

Mathematics 2025, 13(5), 881; https://doi.org/10.3390/math13050881 - 6 Mar 2025

Cited by 5 | Viewed by 866

Abstract

Conformal maps preserve angles and maintain the local shape of geometric structures. The osculating curve plays an important role in analyzing the variations in curvature, providing a detailed understanding of the local geometric properties and the impact of conformal transformations on curves and [...] Read more.

Conformal maps preserve angles and maintain the local shape of geometric structures. The osculating curve plays an important role in analyzing the variations in curvature, providing a detailed understanding of the local geometric properties and the impact of conformal transformations on curves and surfaces. In this paper, we study osculating curves on regular surfaces under conformal transformations. We obtained the conditions required for osculating curves on regular surfaces R and

\tilde{R}

to remain invariant when subjected to a conformal transformation

ψ : R \to \tilde{R}

. The results presented in this paper reveal the specific conditions under which the transformed curve

\tilde{σ} = ψ \circ σ

preserves its osculating properties, depending on whether

\tilde{σ}

is a geodesic, asymptotic, or neither. Furthermore, we analyze these conditions separately for cases with zero and non-zero normal curvatures. We also explore the behavior of these curves along the tangent vector

T_{σ}

and the unit normal vector

P_{σ}

. Full article

(This article belongs to the Special Issue Geometric Topology and Differential Geometry with Applications)

20 pages, 375 KB

Open AccessFeature PaperEditor’s ChoiceArticle

On Error Estimation and Convergence of the Difference Scheme for a Nonlinear Elliptic Equation with an Integral Boundary Condition

by Regimantas Čiupaila, Mifodijus Sapagovas, Kristina Pupalaigė and Gailė Kamilė Šaltenienė

Mathematics 2025, 13(5), 873; https://doi.org/10.3390/math13050873 - 5 Mar 2025

Viewed by 867

Abstract

In this paper, a two-dimensional nonlinear elliptic equation with an integral boundary condition depending on two parameters is investigated. The problem is solved using the finite difference method. The error in the solution is evaluated based on the properties of M-matrices, and herewith [...] Read more.

In this paper, a two-dimensional nonlinear elliptic equation with an integral boundary condition depending on two parameters is investigated. The problem is solved using the finite difference method. The error in the solution is evaluated based on the properties of M-matrices, and herewith the convergence of the difference scheme is proved. The majorant is constructed to estimate the error of the solution of the system of difference equations. Full article

(This article belongs to the Special Issue New Trends in Nonlinear Analysis)

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27 pages, 3808 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Dynamic Modeling of Limit Order Book and Market Maker Strategy Optimization Based on Markov Queue Theory

by Fei Xie, Yang Liu, Changlong Hu and Shenbao Liang

Mathematics 2025, 13(5), 778; https://doi.org/10.3390/math13050778 - 26 Feb 2025

Viewed by 5443

Abstract

In recent years, high-frequency trading has become increasingly popular in financial markets, making the dynamic modeling of the limit book and the optimization of market maker strategies become key topics. However, existing studies often lacked detailed descriptions of order books and failed to [...] Read more.

In recent years, high-frequency trading has become increasingly popular in financial markets, making the dynamic modeling of the limit book and the optimization of market maker strategies become key topics. However, existing studies often lacked detailed descriptions of order books and failed to fully characterize the optimal decisions of market makers in complex market environments, especially in China’s A-share market. Based on Markov queue theory, this paper proposes the dynamic model of the limit order and the optimal strategy of the market maker. The model uses a state transition probability matrix to refine the market diffusion state, order generation, and trading process and incorporates indicators such as optimal quote deviation and restricted order trading probability. Then, the optimal control model is constructed and the reference strategy is derived using the Hamilton–Jacobi–Bellman (HJB) equation. Then, the key parameters are estimated using the high-frequency data of Ping An Bank for a single trading day. In the empirical aspect, the six-month high-frequency trading data of 114 representative stocks in different market states such as the bull market and bear market in China’s A-share market were selected for strategy verification. The results showed that the proposed strategy had robust returns and stable profits in the bull market and that frequent capture of market fluctuations in the bear market can earn relatively high returns while maintaining 50% of the order coverage rate and 66% of the stable order winning rate. Our study used Markov queuing theory to describe the state and price dynamics of the limit order book in detail and used optimization methods to construct and solve the optimal market maker strategy. The empirical aspect broadens the empirical scope of market maker strategies in the Chinese market and studies the stability and effectiveness of market makers in different market states. Full article

(This article belongs to the Section E: Applied Mathematics)

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23 pages, 3224 KB

Open AccessEditor’s ChoiceArticle

AMC: Adaptive Learning Rate Adjustment Based on Model Complexity

by Weiwei Cheng, Rong Pu and Bin Wang

Mathematics 2025, 13(4), 650; https://doi.org/10.3390/math13040650 - 16 Feb 2025

Cited by 4 | Viewed by 2717

Abstract

An optimizer plays a decisive role in the efficiency and effectiveness of model training in deep learning. Although Adam and its variants are widely used, the impact of model complexity on training is not considered, which leads to instability or slow convergence when [...] Read more.

An optimizer plays a decisive role in the efficiency and effectiveness of model training in deep learning. Although Adam and its variants are widely used, the impact of model complexity on training is not considered, which leads to instability or slow convergence when a complex model is trained. To address this issue, we propose an AMC (Adam with Model Complexity) optimizer, which dynamically adjusts the learning rate by incorporating model complexity, thereby improving training stability and convergence speed. AMC uses the Frobenius norm of the model to measure its complexity, automatically decreasing the learning rate of complex models and increasing the learning rate of simple models, thus optimizing the training process. We provide a theoretical analysis to demonstrate the relationship between model complexity and learning rate, as well as the convergence and convergence bounds of AMC. Experiments on multiple benchmark datasets show that, compared to several widely used optimizers, AMC exhibits better stability and faster convergence, especially in the training of complex models. Full article

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17 pages, 337 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Support Vector Machines and Model Selection for Control Chart Pattern Recognition

by Chih-Jen Su, I-Fei Chen, Tzong-Ru Tsai, Tzu-Hsuan Wang and Yuhlong Lio

Mathematics 2025, 13(4), 592; https://doi.org/10.3390/math13040592 - 11 Feb 2025

Cited by 2 | Viewed by 1259

Abstract

Resource-intensiveness often occurs in modern industrial settings; meanwhile, common issues and irregular patterns in production can lead to defects and variations in work-piece dimensions, negatively impacting products and increasing costs. Utilizing traditional process control charts to monitor the process and identify potential anomalies [...] Read more.

Resource-intensiveness often occurs in modern industrial settings; meanwhile, common issues and irregular patterns in production can lead to defects and variations in work-piece dimensions, negatively impacting products and increasing costs. Utilizing traditional process control charts to monitor the process and identify potential anomalies is expensive when intensive resources are needed. To conquer these downsides, algorithms for control chart pattern recognition (CCPR) leverage machine learning models to detect non-normality or normality and ensure product quality is established, and novel approaches that integrate the support vector machine (SVM), random forest (RF), and K-nearest neighbors (KNN) methods with the model selection criterion, named SVM-, RF-, and KNN-CCPR, respectively, are proposed. The three CCPR approaches can save sample resources in the initial process monitoring, improve the weak learner’s ability to recognize non-normal data, and include normality as a special case. Simulation results and case studies show that the proposed SVM-CCPR method outperforms the other two competitors with the highest recognition rate and yields favorable performance for quality control. Full article

(This article belongs to the Special Issue Mathematical Applications in Industrial Engineering)

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44 pages, 1327 KB

Open AccessFeature PaperEditor’s ChoiceArticle

On Spectral Graph Determination

by Igal Sason, Noam Krupnik, Suleiman Hamud and Abraham Berman

Mathematics 2025, 13(4), 549; https://doi.org/10.3390/math13040549 - 7 Feb 2025

Cited by 1 | Viewed by 2595

Abstract

The study of spectral graph determination is a fascinating area of research in spectral graph theory and algebraic combinatorics. This field focuses on examining the spectral characterization of various classes of graphs, developing methods to construct or distinguish cospectral nonisomorphic graphs, and analyzing [...] Read more.

The study of spectral graph determination is a fascinating area of research in spectral graph theory and algebraic combinatorics. This field focuses on examining the spectral characterization of various classes of graphs, developing methods to construct or distinguish cospectral nonisomorphic graphs, and analyzing the conditions under which a graph’s spectrum uniquely determines its structure. This paper presents an overview of both classical and recent advancements in these topics, along with newly obtained proofs of some existing results, which offer additional insights. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 2nd Edition)

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24 pages, 420 KB

Open AccessEditor’s ChoiceArticle

A Group Consensus Measure That Takes into Account the Relative Importance of the Decision-Makers

by József Dombi, Jenő Fáró and Tamás Jónás

Mathematics 2025, 13(3), 526; https://doi.org/10.3390/math13030526 - 5 Feb 2025

Cited by 1 | Viewed by 1151

Abstract

In group decision making, the knowledge, skills, and experience of the decision-makers may not be at the same level. Hence, the need arises to take into account not only the opinion, but also the relative importance of the opinion of each decision-maker. These [...] Read more.

In group decision making, the knowledge, skills, and experience of the decision-makers may not be at the same level. Hence, the need arises to take into account not only the opinion, but also the relative importance of the opinion of each decision-maker. These relative importance values can be treated as weights. In a group decision making situation, it is not only the weighted aggregate output that matters, but also the weighted measure of the group consensus. Noting that weighted group consensus measures have not yet been intensely studied, in this study, based on well-known requirements for non-weighted consensus measures, we define six reasonable requirements for the weighted case. Then, we propose a function family and prove that it satisfies the above requirements for a weighted consensus measure. Hence, the proposed measure can be used in group decision making situations where the decision-makers have various weight values that reflect the relative importance of their opinions. The proposed weighted consensus measure is based on the fuzziness degree of the decumulative distribution function of the input scores, taking into account the weights. Hence, it may be viewed as a weighted adaptation of the so-called fuzziness measure-based consensus measure. The novel weighted consensus measure is determined by a fuzzy entropy function; i.e., this function may be regarded as a generator of the consensus measure. This property of the proposed weighted consensus measure family makes it very versatile and flexible. The nice properties of the proposed weighted consensus measure family are demonstrated by means of concrete numerical examples. Full article

(This article belongs to the Special Issue Advanced Intelligent Algorithms for Decision Making Under Uncertainty)

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50 pages, 9587 KB

Open AccessEditor’s ChoiceArticle

Uncertain Numbers

by Peng Yue

Mathematics 2025, 13(3), 496; https://doi.org/10.3390/math13030496 - 2 Feb 2025

Cited by 9 | Viewed by 3532

Abstract

This work presents a mathematical framework based on uncertain numbers to address the inherent uncertainty in nonlinear systems, a challenge that traditional mathematical frameworks often struggle to fully capture. By establishing five axioms, a formal system of uncertain numbers is developed and embedded [...] Read more.

This work presents a mathematical framework based on uncertain numbers to address the inherent uncertainty in nonlinear systems, a challenge that traditional mathematical frameworks often struggle to fully capture. By establishing five axioms, a formal system of uncertain numbers is developed and embedded within set theory, providing a comprehensive characterization of uncertainty. This framework allows phenomena such as infinity and singularities to be treated as uncertain numbers, offering a mathematically rigorous analytical approach. Subsequently, an algebraic structure for uncertain numbers is constructed, defining fundamental operations such as addition, subtraction, multiplication, and division. The framework is compatible with existing mathematical paradigms, including complex numbers, fuzzy numbers, and probability theory, thereby forming a unified theoretical structure for quantifying and analyzing uncertainty. This advancement not only provides new avenues for research in mathematics and physics but also holds significant practical value, particularly in improving numerical methods to address singularity problems and optimizing nonconvex optimization algorithms. Additionally, the anti-integral-saturation technique, widely applied in control science, is rigorously derived within this framework. These applications highlight the utility and reliability of the uncertain number framework in both theoretical and practical domains. Full article

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19 pages, 389 KB

Open AccessFeature PaperEditor’s ChoiceArticle

On the Existence and Uniqueness of Solutions for Neutral-Type Caputo Fractional Differential Equations with Iterated Delays: Hyers–Ulam–Mittag–Leffler Stability

by Ekaterina Madamlieva and Mihail Konstantinov

Mathematics 2025, 13(3), 484; https://doi.org/10.3390/math13030484 - 31 Jan 2025

Cited by 2 | Viewed by 838

Abstract

This study investigates nonlinear Caputo-type fractional differential equations with iterated delays, focusing on the neutral type. Initially formulated by D. Bainov and the second author of the current paper between 1972 and 1978, these superneutral equations have been extensively studied in scholarly inquiry. [...] Read more.

This study investigates nonlinear Caputo-type fractional differential equations with iterated delays, focusing on the neutral type. Initially formulated by D. Bainov and the second author of the current paper between 1972 and 1978, these superneutral equations have been extensively studied in scholarly inquiry. The present research seeks to reinvigorate interest in such delays within sophisticated frameworks of differential equations, particularly those involving fractional calculus. The primary objectives are to thoroughly examine neutral-type fractional differential equations with iterated delays and provide novel insights into their existence and uniqueness by applying Bielecki’s and Chebyshev’s norms for solution constraints analysis. Additionally, this work establishes Hyers–Ulam–Mittag–Leffler stability for these equations. Full article

(This article belongs to the Special Issue Advances in Differential Dynamical Systems with Applications to Economics and Biology, 3rd Edition)

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29 pages, 8288 KB

Open AccessEditor’s ChoiceArticle

Partial Least Squares Regression for Binary Data

by Laura Vicente-Gonzalez, Elisa Frutos-Bernal and Jose Luis Vicente-Villardon

Mathematics 2025, 13(3), 458; https://doi.org/10.3390/math13030458 - 30 Jan 2025

Cited by 3 | Viewed by 1958

Abstract

Classical Partial Least Squares Regression (PLSR) models were developed primarily for continuous data, allowing dimensionality reduction while preserving relationships between predictors and responses. However, their application to binary data is limited. This study introduces Binary Partial Least Squares Regression (BPLSR), a novel extension [...] Read more.

Classical Partial Least Squares Regression (PLSR) models were developed primarily for continuous data, allowing dimensionality reduction while preserving relationships between predictors and responses. However, their application to binary data is limited. This study introduces Binary Partial Least Squares Regression (BPLSR), a novel extension of the PLSR methodology designed specifically for scenarios involving binary predictors and responses. BPLSR adapts the classical PLSR framework to handle the unique properties of binary datasets. A key feature of this approach is the introduction of a triplot representation that integrates logistic biplots. This visualization tool provides an intuitive interpretation of relationships between individuals and variables from both predictor and response matrices, enhancing the interpretability of binary data analysis. To illustrate the applicability and effectiveness of BPLSR, the method was applied to a real-world dataset of strains of Colletotrichum graminicola, a pathogenic fungus. The results demonstrated the ability of the method to represent binary relationships between predictors and responses, underscoring its potential as a robust analytical tool. This work extends the capabilities of traditional PLSR methods and provides a practical and versatile solution for binary data analysis with broad applications in diverse research areas. Full article

(This article belongs to the Section D1: Probability and Statistics)

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26 pages, 3650 KB

Open AccessEditor’s ChoiceArticle

Geometrical Optics Stability Analysis of Rotating Visco-Diffusive Flows

by Oleg Kirillov

Mathematics 2025, 13(3), 382; https://doi.org/10.3390/math13030382 - 24 Jan 2025

Cited by 1 | Viewed by 1217

Abstract

Geometrical optics stability analysis has proven effective in deriving analytical instability criteria for 3D flows in ideal hydrodynamics and magnetohydrodynamics, encompassing both compressible and incompressible fluids. The method models perturbations as high-frequency wavelets, evolving along fluid trajectories. Detecting local instabilities reduces to solving [...] Read more.

Geometrical optics stability analysis has proven effective in deriving analytical instability criteria for 3D flows in ideal hydrodynamics and magnetohydrodynamics, encompassing both compressible and incompressible fluids. The method models perturbations as high-frequency wavelets, evolving along fluid trajectories. Detecting local instabilities reduces to solving ODEs for the wave vector and amplitude of the wavelet envelope along streamlines, with coefficients derived from the background flow. While viscosity and diffusivity were traditionally regarded as stabilizing factors, recent extensions of the geometrical optics framework have revealed their destabilizing potential in visco-diffusive and multi-diffusive flows. This review highlights these advancements, with a focus on their application to the azimuthal magnetorotational instability in magnetohydrodynamics and the McIntyre instability in lenticular vortices and swirling differentially heated flows. It introduces new analytical instability criteria, applicable across a wide range of Prandtl, Schmidt, and magnetic Prandtl numbers, which still remains beyond the reach of numerical methods in many important physical and industrial applications. Full article

(This article belongs to the Special Issue Numerical Simulation and Methods in Computational Fluid Dynamics)

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17 pages, 343 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Gaussian Process Regression with Soft Equality Constraints

by Didem Kochan and Xiu Yang

Mathematics 2025, 13(3), 353; https://doi.org/10.3390/math13030353 - 22 Jan 2025

Cited by 1 | Viewed by 1379

Abstract

This study introduces a novel Gaussian process (GP) regression framework that probabilistically enforces physical constraints, with a particular focus on equality conditions. The GP model is trained using the quantum-inspired Hamiltonian Monte Carlo (QHMC) algorithm, which efficiently samples from a wide range of [...] Read more.

This study introduces a novel Gaussian process (GP) regression framework that probabilistically enforces physical constraints, with a particular focus on equality conditions. The GP model is trained using the quantum-inspired Hamiltonian Monte Carlo (QHMC) algorithm, which efficiently samples from a wide range of distributions by allowing a particle’s mass matrix to vary according to a probability distribution. By integrating QHMC into the GP regression with probabilistic handling of the constraints, this approach balances the computational cost and accuracy in the resulting GP model, as the probabilistic nature of the method contributes to shorter execution times compared with existing GP-based approaches. Additionally, we introduce an adaptive learning algorithm to optimize the selection of constraint locations to further enhance the flexibility of the method. We demonstrate the effectiveness and robustness of our algorithm on synthetic examples, including 2-dimensional and 10-dimensional GP models under noisy conditions, as well as a practical application involving the reconstruction of a sparsely observed steady-state heat transport problem. The proposed approach reduces the posterior variance in the resulting model, achieving stable and accurate sampling results across all test cases while maintaining computational efficiency. Full article

(This article belongs to the Special Issue Machine Learning and Statistical Learning with Applications)

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38 pages, 9211 KB

Open AccessEditor’s ChoiceArticle

Transfinite Patches for Isogeometric Analysis

by Christopher Provatidis

Mathematics 2025, 13(3), 335; https://doi.org/10.3390/math13030335 - 21 Jan 2025

Cited by 4 | Viewed by 904

Abstract

This paper extends the well-known transfinite interpolation formula, which was developed in the late 1960s by the applied mathematician William Gordon at the premises of General Motors as an extension of the pre-existing Coons interpolation formula. Here, a conjecture is formulated, which claims [...] Read more.

This paper extends the well-known transfinite interpolation formula, which was developed in the late 1960s by the applied mathematician William Gordon at the premises of General Motors as an extension of the pre-existing Coons interpolation formula. Here, a conjecture is formulated, which claims that the meaning of the involved blending functions can be enhanced, such that it includes any linear independent and complete set of functions, including piecewise-linear, trigonometric functions, Bernstein polynomials, B-splines, and NURBS, among others. In this sense, NURBS-based isogeometric analysis and aspects of T-splines may be considered as special cases. Applications are provided to illustrate the accuracy in the interpolation through the L₂ error norm of closed-formed functions prescribed at the nodal points of the transfinite patch, which represent the solution of partial differential equations under boundary conditions of the Dirichlet type. Full article

(This article belongs to the Special Issue Advanced Numerical and Computational Methods for Engineering and Applied Mathematical Problems, 2nd Edition)

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22 pages, 3406 KB

Open AccessEditor’s ChoiceArticle

Design of a Multi-Layer Symmetric Encryption System Using Reversible Cellular Automata

by George Cosmin Stănică and Petre Anghelescu

Mathematics 2025, 13(2), 304; https://doi.org/10.3390/math13020304 - 18 Jan 2025

Cited by 1 | Viewed by 1376

Abstract

The increasing demand for secure and efficient encryption algorithms has intensified the exploration of alternative cryptographic solutions, including biologically inspired systems like cellular automata. This study presents a symmetric block encryption design based on multiple reversible cellular automata (RCAs) that can assure both [...] Read more.

The increasing demand for secure and efficient encryption algorithms has intensified the exploration of alternative cryptographic solutions, including biologically inspired systems like cellular automata. This study presents a symmetric block encryption design based on multiple reversible cellular automata (RCAs) that can assure both computational efficiency and reliable restoration of original data. The encryption key, with a length of 224 bits, is composed of specific rules used by the four distinct RCAs: three with radius-2 neighborhoods and one with a radius-3 neighborhood. By dividing plaintext into 128-bit blocks, the algorithm performs iterative transformations over multiple rounds. Each round includes forward or backward evolution steps, along with dynamically computed shift values and reversible transformations to securely encrypt or decrypt data. The encryption process concludes with an additional layer of security by encrypting the final RCA configurations, further protecting against potential attacks on the encrypted data. Additionally, the 224-bit key length provides robust resistance against brute force attacks. Testing and analysis were performed using a custom-developed software (version 1.0) application, which helped demonstrate the algorithm’s robustness, encryption accuracy, and ability to maintain data integrity. Full article

(This article belongs to the Section E1: Mathematics and Computer Science)

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13 pages, 2779 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Convergence and Dynamics of Schröder’s Method for Zeros of Analytic Functions with Unknown Multiplicity

by Plamena I. Marcheva and Stoil I. Ivanov

Mathematics 2025, 13(2), 275; https://doi.org/10.3390/math13020275 - 16 Jan 2025

Cited by 2 | Viewed by 888

Abstract

In this paper, we investigate the local convergence of Schröder’s method for finding zeros of analytic functions with unknown multiplicity. Thus, we obtain a convergence theorem that provides exact domains of initial points together with error estimates to ensure the Q-quadratic convergence [...] Read more.

In this paper, we investigate the local convergence of Schröder’s method for finding zeros of analytic functions with unknown multiplicity. Thus, we obtain a convergence theorem that provides exact domains of initial points together with error estimates to ensure the Q-quadratic convergence of Schröder’s method right from the first step. A comparison with the famous Newton’s method, based on the convergence and dynamics when it is applied to some polynomial and non-polynomial equations, is also provided. Full article

(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis: 2nd Edition)

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19 pages, 308 KB

Open AccessFeature PaperEditor’s ChoiceArticle

On the Exponential Atom-Bond Connectivity Index of Graphs

by Kinkar Chandra Das

Mathematics 2025, 13(2), 269; https://doi.org/10.3390/math13020269 - 15 Jan 2025

Cited by 4 | Viewed by 1251

Abstract

Several topological indices are possibly the most widely applied graph-based molecular structure descriptors in chemistry and pharmacology. The capacity of topological indices to discriminate is a crucial component of their study. In light of this, the literature has introduced the exponential vertex-degree-based topological [...] Read more.

Several topological indices are possibly the most widely applied graph-based molecular structure descriptors in chemistry and pharmacology. The capacity of topological indices to discriminate is a crucial component of their study. In light of this, the literature has introduced the exponential vertex-degree-based topological index. The exponential atom-bond connectivity index is defined as follows:

e^{ABC} = e^{ABC} (Υ) = \sum_{v_{i} v_{j} \in E (Υ)} e^{\sqrt{\frac{d_{i} + d_{j} - 2}{d_{i} d_{j}}}},

where

d_{i}

is the degree of the vertex

v_{i}

in

Υ

. In this paper, we prove that the double star

D S_{n - 3, 1}

is the second maximal graph with respect to the

e^{ABC}

index of trees of order n. We give an upper bound on

e^{ABC}

of unicyclic graphs of order n and characterize the maximal graphs. The graph

K_{1} \lor (P_{3} \cup (n - 4) K_{1})

gives the maximal graph with respect to the

e^{ABC}

index of bicyclic graphs of order n. We present several relations between

e^{ABC} (Υ)

and

A B C (Υ)

of graph

Υ

. Finally, we provide a conclusion summarizing our findings and discuss potential directions for future research. Full article

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24 pages, 344 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Localization and Flatness in Quantale Theory

by George Georgescu

Mathematics 2025, 13(2), 227; https://doi.org/10.3390/math13020227 - 11 Jan 2025

Viewed by 857

Abstract

The study of flat ring morphisms is an important theme in commutative algebra. The purpose of this article is to develop an abstract theory of flatness in the framework of coherent quantales. The first question we must address is the definition of a [...] Read more.

The study of flat ring morphisms is an important theme in commutative algebra. The purpose of this article is to develop an abstract theory of flatness in the framework of coherent quantales. The first question we must address is the definition of a notion of “flat quantale morphism” as an abstraction of flat ring morphisms. For this, we start from a characterization of the flat ring morphism in terms of the ideal residuation theory. The flat coherent quantale morphism is studied in relation to the localization of coherent quantales. The quantale generalizations of some classical theorems from the flat ring morphisms theory are proved. The Going-down and Going-up properties are then studied in connection with localization theory and flat quantale morphisms. As an application, characterizations of zero-dimensional coherent quantales are obtained, formulated in terms of Going-down, Going-up, and localization. We also prove two characterization theorems for the coherent quantales of dimension at most one. The results of the paper can be applied both in the theory of commutative rings and to other algebraic structures: F-rings, semirings, bounded distributive lattices, commutative monoids, etc. Full article

(This article belongs to the Special Issue Functional Analysis, Topology and Quantum Mechanics, 3rd Edition)

32 pages, 2819 KB

Open AccessEditor’s ChoiceArticle

Disentangling Sources of Multifractality in Time Series

by Robert Kluszczyński, Stanisław Drożdż, Jarosław Kwapień, Tomasz Stanisz and Marcin Wątorek

Mathematics 2025, 13(2), 205; https://doi.org/10.3390/math13020205 - 9 Jan 2025

Cited by 7 | Viewed by 1222

Abstract

This contribution addresses the question commonly asked in the scientific literature about the sources of multifractality in time series. Two primary sources are typically considered. These are temporal correlations and heavy tails in the distribution of fluctuations. Most often, they are treated as [...] Read more.

This contribution addresses the question commonly asked in the scientific literature about the sources of multifractality in time series. Two primary sources are typically considered. These are temporal correlations and heavy tails in the distribution of fluctuations. Most often, they are treated as two independent components, while true multifractality cannot occur without temporal correlations. The distributions of fluctuations affect the span of the multifractal spectrum only when correlations are present. These issues are illustrated here using series generated by several model mathematical cascades, which by design build correlations into these series. The thickness of the tails of fluctuations in such series is then governed by an appropriate procedure of adjusting them to q-Gaussian distributions, and q is treated as a variable parameter that, while preserving correlations, allows for tuning these distributions to the desired functional form. Multifractal detrended fluctuation analysis (MFDFA), as the most commonly used practical method for quantifying multifractality, is then used to identify the influence of the thickness of the fluctuation tails in the presence of temporal correlations on the width of multifractal spectra. The obtained results point to the Gaussian distribution, so

q = 1

, as the appropriate reference distribution to evaluate the contribution of fatter tails to the width of multifractal spectra. An appropriate procedure is presented to make such estimates. Full article

(This article belongs to the Special Issue Recent Advances in Time Series Analysis)

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15 pages, 885 KB

Open AccessEditor’s ChoiceArticle

Estimating the Relative Risks of Spatial Clusters Using a Predictor–Corrector Method

by Majid Bani-Yaghoub, Kamel Rekab, Julia Pluta and Said Tabharit

Mathematics 2025, 13(2), 180; https://doi.org/10.3390/math13020180 - 7 Jan 2025

Cited by 3 | Viewed by 1088

Abstract

Spatial, temporal, and space–time scan statistics can be used for geographical surveillance, identifying temporal and spatial patterns, and detecting outliers. While statistical cluster analysis is a valuable tool for identifying patterns, optimizing resource allocation, and supporting decision-making, accurately predicting future spatial clusters remains [...] Read more.

Spatial, temporal, and space–time scan statistics can be used for geographical surveillance, identifying temporal and spatial patterns, and detecting outliers. While statistical cluster analysis is a valuable tool for identifying patterns, optimizing resource allocation, and supporting decision-making, accurately predicting future spatial clusters remains a significant challenge. Given the known relative risks of spatial clusters over the past k time intervals, the main objective of the present study is to predict the relative risks for the subsequent interval,

k + 1

. Building on our prior research, we propose a predictive Markov chain model with an embedded corrector component. This corrector utilizes either multiple linear regression or an exponential smoothing method, selecting the one that minimizes the relative distance between the observed and predicted values in the k-th interval. To test the proposed method, we first calculated the relative risks of statistically significant spatial clusters of COVID-19 mortality in the U.S. over seven time intervals from May 2020 to March 2023. Then, for each time interval, we selected the top 25 clusters with the highest relative risks and iteratively predicted the relative risks of clusters from intervals three to seven. The predictive accuracies ranged from moderate to high, indicating the potential applicability of this method for predictive disease analytic and future pandemic preparedness. Full article

(This article belongs to the Section E: Applied Mathematics)

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