Mathematical Perspective Research Articles

Addressing imperfect symmetry: A novel symmetry-learning actor-critic extension

Symmetry, a fundamental concept to understand our environment, often oversimplifies reality from a mathematical perspective. Humans are a prime example, deviating from perfect symmetry in terms of appearance and cognitive biases (e.g. having a dominant hand). Nevertheless, our brain can easily overcome these imperfections and efficiently adapt to symmetrical tasks. The driving motivation behind this work lies in capturing this ability through reinforcement learning. To this end, we introduce Adaptive Symmetry Learning (ASL) — a model-minimization actor-critic extension that addresses incomplete or inexact symmetry descriptions by adapting itself during the learning process. ASL consists of a symmetry fitting component and a modular loss function that enforces a common symmetric relation across all states while adapting to the learned policy. The performance of ASL is compared to existing symmetry-enhanced methods in a case study involving a four-legged ant model for multidirectional locomotion tasks. The results show that ASL can recover from large perturbations and generalize knowledge to hidden symmetric states. It achieves comparable or better performance than alternative methods in most scenarios, making it a valuable approach for leveraging model symmetry while compensating for inherent perturbations.

Multi-innovation adaptive Kalman filter algorithm for estimating the SOC of lithium-ion batteries based on singular value decomposition and Schmidt orthogonal transformation

The state of charge (SOC) of lithium battery is a key parameter for effective management of battery management systems (BMS). To address the problems of low precision, complex computation and poor robustness of traditional charging state estimation methods, an enhanced algorithm based on Unscented Kalman filter (UKF) is proposed. The singular value decomposition (SVD) method is adopted to ensure the normal operation of the Unscented Kalman Filter (UKF) algorithm even when the matrix P lacks positive semi-definiteness from a mathematical perspective. This enhancement significantly improves the theoretical robustness of UKF. Schmidt orthogonal transformation is concurrently used to reduce the computational complexity in the sampling point selection process, and the multi-innovation theory is combined with adaptive noise control to further improve the accuracy of SOC estimation. The algorithm is validated using the Urban Dynamometer Driving Schedule (UDDS) condition. The simulation results are excited using Singular value decomposition-multi-innovation adaptive Schmidt orthogonal unscented Kalman filter method (SVD-MIASOUKF). 0.95 % and 1.29 % of maximum errors are obtained at 25 °C and −10 °C, while the maximum errors are 2.46 %–2.99 % using SVD-UKF and UKF. The proposed approach shows faster convergence speed and higher estimation accuracy in comparison to traditional algorithms.

Enhancing Adversarial Robustness through Stable Adversarial Training

Deep neural network models are vulnerable to attacks from adversarial methods, such as gradient attacks. Evening small perturbations can cause significant differences in their predictions. Adversarial training (AT) aims to improve the model’s adversarial robustness against gradient attacks by generating adversarial samples and optimizing the adversarial training objective function of the model. Existing methods mainly focus on improving robust accuracy, balancing natural and robust accuracy and suppressing robust overfitting. They rarely consider the AT problem from the characteristics of deep neural networks themselves, such as the stability properties under certain conditions. From a mathematical perspective, deep neural networks with stable training processes may have a better ability to suppress overfitting, as their training process is smoother and avoids sudden drops in performance. We provide a proof of the existence of Ulam stability for deep neural networks. Ulam stability not only determines the existence of the solution for an operator inequality, but it also provides an error bound between the exact and approximate solutions. The feature subspace of a deep neural network with Ulam stability can be accurately characterized and constrained by a function with special properties and a controlled error boundary constant. This restricted feature subspace leads to a more stable training process. Based on these properties, we propose an adversarial training framework called Ulam stability adversarial training (US-AT). This framework can incorporate different Ulam stability conditions and benchmark AT models, optimize the construction of the optimal feature subspace, and consistently improve the model’s robustness and training stability. US-AT is simple and easy to use, and it can be easily integrated with existing multi-class AT models, such as GradAlign and TRADES. Experimental results show that US-AT methods can consistently improve the robust accuracy and training stability of benchmark models.

A parametric similarity measure for neutrosophic set and its applications in energy production

As a useful tool for managing ambiguous and inconsistent data, the Single Value Neutrosophic Set (SVNSs) is an extension of both Fuzzy Sets (FSs) and Intuitionistic Fuzzy Sets (IFSs). In the field of information theory, metrics like similarity, entropy, and distance are important. Although a number of entropy measures for SVNSs have been put forth and used in real-world situations, both academic research and real-world applications have pointed out certain drawbacks. Additionally, the Similarity Measures (SMs) is a useful instrument for determining how similar any two fuzzy values are to one another. The distance between the values allows the current SMs to evaluate the similarity. However, due to a few characteristics and intricate value operations, there are irrational and nonsensical cases. To deal with these preposterous cases, this paper proposed a parametric similarity measure in view of three parameters m1,m2,m3 in which decision makers can obtain the appropriate SMs by changing parameters with different decision styles. Furthermore, we analyze some existing SMs from a mathematical perspective and demonstrate the success of the proposed SMs using mathematical models. Ultimately, we apply the suggested SMs to resolve the Multi-Attribute Decision-Making (MADM) problems. We learn from the correlation and analysis that the suggested SM outperforms certain other SMs that are based on the SVNSs.

의미론적 정합성 관점에서 분수의 연산 단원 분석: 의미 관계와 수량 유형을 중심으로

Mathematical concepts are derived from real-world situations, and their construction and application can be facilitated when real-world meanings and mathematical meanings are aligned. According to educational psychology research, the depth of interpretation and use of operations depend on whether the semantic relations between the objects presented in the problem situation and the potential mathematical relations are consistent. This study analyzed the semantic relations between objects in problem situations and the semantic alignment between the quantity types of objects and visual models in the fraction operation units of elementary school mathematics textbooks. The analysis revealed that, in problem situations, symmetric relationships were predominantly presented in fraction addition, subset-set relationships in fraction subtraction, and asymmetric relationships in fraction multiplication and division. Regarding quantity types, continuous quantities were presented more often than discrete ones, and in most cases, the quantity type of the object matched that of the visual model, though a few did not. Based on these findings, implications were drawn for selecting problem situations and instructional models that align with mathematical meanings and support student understanding from the perspective of mathematics learning psychology.

A mathematical perspective on Romanisation: Modelling the Roman road activation process in ancient Tunisia.

Romanisation is a multi-faceted historical phenomenon with profound and lasting cultural impact on the ancient world. In the modern-day territory of Tunisia, this is particularly manifest during the first four centuries AD, under the reign of the Roman Empire. We derive a reduced, operational concept of Romanisation as a cultural diffusion process that is observable in the archaeological remains of the Roman era settlement system. We then introduce a novel mathematical model that computes spatio-temporal approximations for the Romanisation of the settlement system. The model is based on the concept of temporal road activation and makes minimal assumptions regarding input data quality. The results of our study contribute to the understanding of the time dynamics of the region's road network, under the influence of Romanisation. Our model can be applied in similar archaeological research scenarios, to generate spatio-temporal backbones for the analysis of otherwise intractably complex social processes.

Using academic mathematical knowledge when working on interface tasks–analyses of pre-service teachers’ arguments about rotationally symmetric figures

AbstractSpecial tasks for pre-service teachers (PSTs) in university mathematics courses (“interface tasks”) are a common innovation in recent years to overcome the second discontinuity. By this, we mean tasks that are situated by typical everyday challenges of mathematics teaching and in which PSTs must use their mathematical knowledge and skills in a professionally relevant way. In this paper, we analyze answers that PSTs have created to an interface task on symmetry. The PSTs were asked to clarify a student’s question from a mathematical perspective and then give a suitable elementarized answer. We situate these two steps theoretically and reconstruct the mathematical reasoning in PSTs' answers. Through qualitative content analysis, we examined how PSTs justify figures' symmetries from a university mathematics perspective and when responding to the fictitious student. The scenario of a student questioning the existence of 100° rotationally symmetrical figures elicited rich and varied responses, proving suitable for an interface task. We compared PSTs' reasoning related to mathematical clarification with the reasoning related to elementarization. In many cases, this revealed a productive use of course content. An interesting result is that there is no uniform picture as to whether the arguments are more detailed in the mathematical clarification or in the elementarization.

Comparative Analysis of Manifold Learning-Based Dimension Reduction Methods: A Mathematical Perspective

Manifold learning-based approaches have emerged as prominent techniques for dimensionality reduction. Among these methods, t-Distributed Stochastic Neighbor Embedding (t-SNE) and Uniform Manifold Approximation and Projection (UMAP) stand out as two of the most widely used and effective approaches. While both methods share similar underlying procedures, empirical observations indicate two distinctive properties: global data structure preservation and computational efficiency. However, the underlying mathematical principles behind these distinctions remain elusive. To address this gap, this study presents a comparative analysis of the subprocesses involved in these methods, aiming to elucidate the mathematical mechanisms underlying the observed distinctions. By meticulously examining the equation formulations, the mathematical mechanisms contributing to global data structure preservation and computational efficiency are elucidated. To validate the theoretical analysis, data are collected through a laboratory experiment, and an open-source dataset is utilized for validation across different datasets. The consistent alignment of results obtained from both balanced and unbalanced datasets robustly confirms the study’s findings. The insights gained from this study provide a deeper understanding of the mathematical underpinnings of t-SNE and UMAP, enabling more informed and effective use of these dimensionality reduction techniques in various applications, such as anomaly detection, natural language processing, and bioinformatics.

The most important thing: Wittgenstein, engineering, and the foundations of mathematics

ABSTRACT This paper revisits Wittgenstein’s heavily criticized claims about the admissibility of inconsistencies in mathematics. It argues from the perspective of mathematics as a tool and combines material from the history and practice of engineering that makes Wittgenstein’s claims about contradiction and inconsistency look much more plausible. Against this background, the paper interprets passages from Wittgenstein, including his exchange with Alan Turing where he highlights that basic laws of thought are at issue and that reflecting on them would be “the most important thing” he has talked about.

Fitting Epidemic Models to Data: A Tutorial in Memory of Fred Brauer.

Fred Brauer was an eminent mathematician who studied dynamical systems, especially differential equations. He made many contributions to mathematical epidemiology, a field that is strongly connected to data, but he always chose to avoid data analysis. Nevertheless, he recognized that fitting models to data is usually necessary when attempting to apply infectious disease transmission models to real public health problems. He was curious to know how one goes about fitting dynamical models to data, and why it can be hard. Initially in response to Fred's questions, we developed a user-friendly R package, fitode, that facilitates fitting ordinary differential equations to observed time series. Here, we use this package to provide a brief tutorial introduction to fitting compartmental epidemic models to a single observed time series. We assume that, like Fred, the reader is familiar with dynamical systems from a mathematical perspective, but has limited experience with statistical methodology or optimization techniques.

Well-posedness of an evaporation model for a spherical droplet exposed to an air flow

In this paper, we address the well-posedness of an evaporation model for a spherical liquid droplet taking into account the convective impact of an air flow in the ambient gas phase. From a mathematical perspective, we are dealing with a coupled ODE–PDE system for the droplet radius, the temperature distribution, and the vapor concentration. The nonlinear coupling arises from the evaporation rate modeled by the Hertz–Knudsen equation. Under physically meaningful assumptions, we prove existence and uniqueness of a weak solution until the droplet has evaporated completely. Numerical simulations are performed to illustrate how different air flows affect the evaporation process.

Dynamical analysis of a stochastic maize streak virus epidemic model with logarithmic Ornstein-Uhlenbeck process.

To describe the transmission dynamics of maize streak virus infection, in the paper, we first formulate a stochastic maize streak virus infection model, in which the stochastic fluctuations are depicted by a logarithmic Ornstein-Uhlenbeck process. This approach is reasonable to simulate the random impacts of main parameters both from the biological significance and the mathematical perspective. Then we investigate the detailed dynamics of the stochastic system, including the existence and uniqueness of the global solution, the existence of a stationary distribution, the exponential extinction of the infected maize and infected leafhopper vector. Especially, by solving the five-dimensional algebraic equations corresponding to the stochastic system, we obtain the specific expression of the probability density function near the quasi-endemic equilibrium of the stochastic system, which provides valuable insights into the stationary distribution. Finally, the model is discretized using the Milstein higher-order numerical method to illustrate our theoretical results numerically. Our findings provide a groundwork for better methods of preventing the spread of this type of virus.

An Overview of Flood Management from a Mathematical Perspective

Floods are among the world’s most devastating natural disasters, resulting in loss of life, damage to property, and widespread infrastructure disruptions that impact millions. Flood modelling is a crucial tool for forecasting and effectively mitigating the consequences of such disasters. This study offers an overview of flood management strategies such as structural and non-structural strategies, and the current state of flood modelling, the challenges it faces, and potential future advancements. The scope of flood modelling encompasses various approaches, such as hydrological and hydraulic models, numerical simulations, rainfall-runoff analysis, remote sensing and geographic information systems (GIS), computational intelligence and robotics. The assessment delves into some of the merits and demerits of different models used to forecast the trajectory and consequences of flood events. This study explores the potential avenues for progress and innovation within the realm of flood modelling, including the integration of modern technology and multifaceted models. To enhance the control of flood hazards and reduce the societal influence of floods, the report underscores the imperative need for continuous research in the field of flood management through conventional and mathematical modelling.

Darboux and generalized Darboux transformations for the fractional integrable derivative nonlinear Schrödinger equation

Analytical methods provide crucial mathematical insights into the stable solitary waves hidden in nonlinear phenomena. The nonlinear Schrödinger (NLS) equation is one of the most important typical integrable soliton models. From a mathematical perspective, the essence of the celebrated derivative NLS (DNLS) equation’s difference from the classical NLS equation lies in its cubic potential being differentiated once by the spatial variable and multiplied by the imaginary unit, which leads to the former having some characteristics that the latter cannot have. This paper extends the DNLS equation to the fractional integrable case with conformable derivative operators, and uses Darboux transformations (DTs) and generalized DT (GDT) to solve it exactly. Specifically, Lax pairs generating the fractional DNLS equation are first given. Based on the given Lax pairs, then the n-fold DTs and GDT for the fractional DNLS equation are derived. Some special exact solutions of the fractional DNLS equation are further obtained by employing the derived n-fold DTs and GDT. Finally, several novel space–time structures and dynamical evolutions of the obtained exact solutions are analyzed. This paper reveals through the DT and GDT methods that the double power-law fractional orders in the exact solutions of the fractional DNLS equation can be used to dominate the variable velocity propagation and anomalous diffusion in fractional dimensional media at different geometric scales.

Current State, Challenges, and Opportunities in Genome-Scale Resource Allocation Models: A Mathematical Perspective.

Stoichiometric genome-scale metabolic models (generally abbreviated GSM, GSMM, or GEM) have had many applications in exploring phenotypes and guiding metabolic engineering interventions. Nevertheless, these models and predictions thereof can become limited as they do not directly account for protein cost, enzyme kinetics, and cell surface or volume proteome limitations. Lack of such mechanistic detail could lead to overly optimistic predictions and engineered strains. Initial efforts to correct these deficiencies were by the application of precursor tools for GSMs, such as flux balance analysis with molecular crowding. In the past decade, several frameworks have been introduced to incorporate proteome-related limitations using a genome-scale stoichiometric model as the reconstruction basis, which herein are called resource allocation models (RAMs). This review provides a broad overview of representative or commonly used existing RAM frameworks. This review discusses increasingly complex models, beginning with stoichiometric models to precursor to RAM frameworks to existing RAM frameworks. RAM frameworks are broadly divided into two categories: coarse-grained and fine-grained, with different strengths and challenges. Discussion includes pinpointing their utility, data needs, highlighting framework strengths and limitations, and appropriateness to various research endeavors, largely through contrasting their mathematical frameworks. Finally, promising future applications of RAMs are discussed.

The use of an imperfect vaccination and awareness campaign in the control of antibiotic resistant gonorrhoea infection: A mathematical modelling perspective

The multi drug-resistant Neisseria gonorrhoeae has been classified by the World Health Organisation (WHO) as a high-priority global public health problem. This underlines the need for better understanding of the transmission dynamics and proposing an optimal intervention strategy to control the disease. In this article, a deterministic mathematical model for the transmission dynamics of gonorrhoea as an antibiotic resistant disease in a population with an imperfect vaccination is proposed and analysed. The model incorporates the classes of vaccinated individuals and individuals equipped with self protection interventions to reduce antibiotic resistant cases. The threshold parameter R0, the basic reproduction number, for the analysis of the model is calculated. In the given setting, the model exhibits a backward bifurcation for R0<1. However, if the efficacy of the vaccine is 100% without a waning effect, the model is shown to be without a backward bifurcation and the disease-free equilibrium is globally asymptotically stable whenever R0<1. The global sensitivity analysis of the model to variations in parameter values is also performed to determine the most influential parameters on the disease transmission. Moreover, the optimal control analysis of the full model is presented and the optimal intervention strategies are proposed. The proposed intervention strategies are shown to be able to control the disease within a relatively shorter period of time. Finally, numerical experiments are presented to support the theoretical analysis of the model.

Hybrid spatial network disintegration strategy considering geographic and topological information

Abstract When considering disintegration of spatial networks, the topological relationships between nodes and their geographical positions are taken into account. In the case of regional attacks, the targeted nodes are determined based on the size of the region and the geographical relationships between nodes. In this paper, a new metric is proposed on top of region centrality (RC) to describe the importance of nodes within local regions in spatial networks. We define the weighted sum of the number of connections between nodes within a local region and the number of connections between nodes within the local region and nodes outside the local region as local connectivity. This metric integrates topological information with geographical information. From a mathematical perspective, RC is a special case of local connectivity. In addition, we propose a hybrid spatial network disintegration strategy based on two metrics, namely local connectivity and RC. From experimental results, our strategy outperforms traditional approaches and demonstrates a stronger ability to completely disintegrate the network.

Dancing with math: Using Kleins quartic for music generation

Studying music from a mathematical perspective is often based on the notions of symmetry and topological space. A group G acting on some set S of musical objects in a meaningful way is seen as a symmetry group in music. This can be used in analysing motif development and chord progressions. In neo-Riemannian analysis, one has three principal chord transformations that generate a group GD_12, acting on the set S of 24 major and minor triads. Moreover, this theory is visualized through a simplicial complex whose underlying space is a topological torus. In this paper, we first introduce various symmetry groups and graphic presentations in music theory. We then propose a way of doing neo-Riemannian analysis on Kleins quartic, which is a genus 3 surface instead of a torus, realizing an idea of John Baez. Finally, we use our theory to perform a harmonic analysis of The Imperial March from Star Wars.

A Role of Effector CD T Cells Against Circulating Tumor Cells Cloaked with Platelets: Insights from a Mathematical Model.

Cancer metastasis accounts for a majority of cancer-related deaths worldwide. Metastasis occurs when the primary tumor sheds cells into the blood and lymphatic circulation, thereby becoming circulating tumor cells (CTCs) that transverse through the circulatory system, extravasate the circulation and establish a secondary distant tumor. Accumulating evidence suggests that circulating effector CD T cells are able to recognize and attack arrested or extravasating CTCs, but this important antitumoral effect remains largely undefined. Recent studies highlighted the supporting role of activated platelets in CTCs's extravasation from the bloodstream, contributing to metastatic progression. In this work, a simple mathematical model describes how the primary tumor, CTCs, activated platelets and effector CD T cells participate in metastasis. The stability analysis reveals that for early dissemination of CTCs, effector CD T cells can present or keep secondary metastatic tumor burden at low equilibrium state. In contrast, for late dissemination of CTCs, effector CD T cells are unlikely to inhibit secondary tumor growth. Moreover, global sensitivity analysis demonstrates that the rate of the primary tumor growth, intravascular CTC proliferation, as well as the CD T cell proliferation, strongly affects the number of the secondary tumor cells. Additionally, model simulations indicate that an increase in CTC proliferation greatly contributes to tumor metastasis. Our simulations further illustrate that the higher the number of activated platelets on CTCs, the higher the probability of secondary tumor establishment. Intriguingly, from a mathematical immunology perspective, our simulations indicate that if the rate of effector CD T cell proliferation is high, then the secondary tumor formation can be considerably delayed, providing a window for adjuvant tumor control strategies. Collectively, our results suggest that the earlier the effector CD T cell response is enhanced the higher is the probability of preventing or delaying secondary tumor metastases.

Stability analysis of the stochastic Grey‐Scott model using spectral method

The Grey‐Scott model is a prominent reaction–diffusion system that has sustain consequential recent research about patterning creation in the reaction–diffusion systems. The present research work, explore the stability analysis of a proposed stochastic Grey‐Scott model, which recommend the randomness or unpredictability into the complex system. To take into account the different variations in reactant concentrations profiles, the stochasticity is modeled using the additive white noise terms. To investigate the stability characteristics of the stochastic Grey‐Scott model, we use the spectral collocation technique. Such mathematical perspective sheds light on the different system's behavior in the addition of white noise, and also it may have significance for comprehending processes in the real world that are controlled by comparable dynamics.