Mathematical Challenges Research Articles

Numerical Methods and Computation in Financial Mathematics: Solving PDEs, Monte Carlo Simulations, and Machine Learning Applications

Abstract. This paper delves into three fundamental numerical methods and computational techniques in financial mathematics: Finite Difference Methods (FDM), Monte Carlo Simulations (MCS), and Machine Learning (ML) applications. Finite Difference Methods are widely utilized for solving partial differential equations (PDEs) in option pricing, with various schemes offering different stability and convergence properties. Monte Carlo Simulations provide a powerful approach for pricing complex derivatives and risk management, addressing the challenges of high-dimensionality and computational complexity. Machine Learning has revolutionized predictive modeling in finance, enabling sophisticated analysis of large datasets to uncover hidden patterns and enhance trading strategies. Through a detailed examination of these methods, including specific examples and data, this paper highlights their theoretical foundations, practical implementations, and the advancements they bring to computational finance. By bridging theoretical approaches with practical applications, we aim to offer insights into the future directions and challenges in financial mathematics.

A gentle introduction to gradient-based optimization and variational inequalities for machine learning

Abstract The rapid progress in machine learning in recent years has been based on a highly productive connection to gradient-based optimization. Further progress hinges in part on a shift in focus from pattern recognition to decision-making and multi-agent problems. In these broader settings, new mathematical challenges emerge that involve equilibria and game theory instead of optima. Gradient-based methods remain essential—given the high dimensionality and large scale of machine-learning problems—but simple gradient descent is no longer the point of departure for algorithm design. We provide a gentle introduction to a broader framework for gradient-based algorithms in machine learning, beginning with saddle points and monotone games, and proceeding to general variational inequalities. While we provide convergence proofs for several of the algorithms that we present, our main focus is that of providing motivation and intuition.

Mathematical Challenges for the Theory of Hyperbolic Balance Laws in Fluid Mechanics: Complexity, Scales, Randomness

AbstractUnderstanding the dynamics of hyperbolic balance laws is of paramount interest in the realm of fluid mechanics. Nevertheless, fundamental questions on the analysis and the numerics for distinctive hyperbolic features related to turbulent flow motion remain vastly open. Recent progress on the mathematical side reveals novel routes to face these concerns. This includes findings about the failure of the entropy principle to ensure uniqueness, the use of structure-preserving concepts in high-order numerical methods, and the advent of tailored probabilistic approaches. Whereas each of these three directions on hyperbolic modelling are of completely different origin they are all linked to small- or subscale features in the solutions which are either enhanced or depleted by the hyperbolic nonlinearity. Thus, any progress in the field might contribute to a deeper understanding of turbulent flow motion on the basis of the continuum-scale mathematical models. We present an overview on the mathematical state-of-the-art in the field and relate it to the scientific work in the DFG Priority Research Programme 2410. As such, the survey is not necessarily targeting at readers with comprehensive knowledge on hyperbolic balance laws but instead aims at a general audience of reseachers which are interested to gain an overview on the field and associated challenges in fluid mechanics.

New trends in didactic research in university mathematics education

AbstractRecent research in university mathematics education has moved beyond the traditional focus on the transition from secondary to tertiary education and students' understanding of introductory courses such as pre-calculus and calculus. There is growing interest in the challenges students face as they move into more advanced mathematics courses that require a shift toward formal reasoning, proof, modeling, and problem-solving skills. This survey paper explores emerging trends and innovations in the field, focusing on three key areas: innovations in teaching and learning advanced mathematical topics, transitions between different levels and contexts of mathematics education, and the role of proof and proving in advanced university mathematics. The survey reflects the evolving landscape of mathematics education research and addresses the theoretical and practical challenges of teaching and learning advanced mathematics across various contexts.

From Play to Understanding: Large Language Models in Logic and Spatial Reasoning Coloring Activities for Children

Visual thinking leverages spatial mechanisms in animals for navigation and reasoning. Therefore, given the challenge of abstract mathematics and logic, spatial reasoning-based teaching strategies can be highly effective. Our previous research verified that innovative box-and-ball coloring activities help teach elementary school students complex notions like quantifiers, logical connectors, and dynamic systems. However, given the richness of the activities, correction is slow, error-prone, and demands high attention and cognitive load from the teacher. Moreover, feedback to the teacher should be immediate. Thus, we propose to provide the teacher with real-time help with LLMs. We explored various prompting techniques with and without context—Zero-Shot, Few-Shot, Chain of Thought, Visualization of Thought, Self-Consistency, logicLM, and emotional —to test GPT-4o’s visual, logical, and correction capabilities. We obtained that Visualization of Thought and Self-Consistency techniques enabled GPT-4o to correctly evaluate 90% of the logical–spatial problems that we tested. Additionally, we propose a novel prompt combining some of these techniques that achieved 100% accuracy on a testing sample, excelling in spatial problems and enhancing logical reasoning.

Development of Interactive Learning Video Media to Improve Mathematical Representation Ability on Data Presentation Material

One of the main challenges in learning mathematics is overcoming students' difficulties in understanding abstract mathematical material and the lack of engaging and interactive teaching methods. The aim of this research is to develop interactive learning videos that are valid, effective, and practical. This Research and Development (R&D) study uses the ADDIE model. The research subjects were 15 students of class VII A MTsS Ash-Shalihin.. The results of the study show that the interactive learning video is highly valid with a score of 3.7 from expert evaluations. Teacher responses received a score of 3.67, which is categorized as very practical, while student responses received a score of 3.34, indicating that the video is very effective. The t-test and N-Gain test showed a value of 0.67, indicating an improvement in students' mathematical representation abilities in the moderate category. Thus, the learning video has proven to be effective.

Nonlinear Optics: Physics, Analysis, and Numerics

When high-intensity electromagnetic waves at optical frequencies interact with solids and/or nanostructures the materials’ response cannot anymore be described via simple linear relations. The resulting science of nonlinear optics has recently witnessed exiting developments that have brought to the fore numerous mathematical challenges that need to be addressed in order to fully exploit the opportunities that result from these developments. The mathematical modeling involves a system of partial differential equations where the Maxwell equations are coupled to evolution equations of the materials and their response to electromagnetic fields. Typically, the full coupled systems are quite complicated or even intractable so that the derivation, the analysis, and the numerical treatment of simplified effective models is often indispensable. In turn, this requires the close cooperation between researchers from theoretical physics and analysis/numerics in order to push forward the field on nonlinear optics.

Multiscale computational analysis of the steady fluid flow through a lymph node.

Lymph Nodes (LNs) are crucial to the immune and lymphatic systems, filtering harmful substances and regulating lymph transport. LNs consist of a lymphoid compartment (LC) that forms a porous bulk region, and a subcapsular sinus (SCS), which is a free-fluid region. Mathematical and mechanical challenges arise in understanding lymph flow dynamics. The highly vascularized lymph node connects the lymphatic and blood systems, emphasizing its essential role in maintaining the fluid balance in the body. In this work, we describe a mathematical model in a steady setting to describe the lymph transport in a lymph node. We couple the fluid flow in the SCS governed by an incompressible Stokes equation with the fluid flow in LC, described by a model obtained by means of asymptotic homogenisation technique, taking into account the multiscale nature of the node and the fluid exchange with the blood vessels inside it. We solve this model using numerical simulations and we analyze the lymph transport inside the node to elucidate its regulatory mechanisms and significance. Our results highlight the crucial role of the microstructure of the lymph node in regularising its fluid balance. These results can pave the way to a better understanding of the mechanisms underlying the lymph node's multiscale functionalities which can be significantly affected by specific physiological and pathological conditions, such as those characterising malignant tissues.

Shape and topology optimization method with generalized topological derivatives

This paper introduces a novel method for shape and topology optimization based on a generalized approach for evaluating topological derivatives, which are essential for the integration of shape and topology optimization. Traditionally, evaluating these derivatives presents significant mathematical challenges due to the discontinuity introduced by the insertion of a hole within the domain of interest. To overcome this issue, the study employs Helmholtz-type partial differential equations (PDEs) to construct a filtered objective functional. This approach ensures differentiability across the material and void phases and continuity over the fixed design domain while maintaining the same evaluation value as the original objective functional. By considering differentiability, continuity conditions, and the relationship between shape and topological derivatives during asymptotic analysis, generalized topological derivatives are obtained through established mathematical procedures. These topological derivatives exhibit a direct correlation with the PDE solutions and demonstrate satisfactory smoothness, thereby facilitating refined shapes in optimization strategies. Furthermore, an effective shape update algorithm is proposed, which directly integrates topological derivatives into structural optimization problems, simplifying their implementation and improving efficiency. Finally, the efficacy of the proposed methodology is demonstrated through its application to various optimal design problems, including stiffness maximization, compliant mechanisms, and eigenfrequency maximization. Verification results further highlight its potential to enhance existing methods for addressing more practical and complex optimization challenges.

Free vibration of doubly-curved panels reinforced by carbon nanotubes: New analytic solutions under non-Lévy-type boundary conditions

Existing analytic solutions for the free vibration of functionally graded carbon nanotube reinforced doubly-curved panels primarily address the cases with two parallel simply supported boundaries, known as Lévy-type boundary conditions (BCs). However, doubly-curved panels with non-Lévy-type BCs are more commonly encountered in practical engineering applications, yet their analytic solutions are rarely available due to significant mathematical challenges. This gap motivates us to develop new analytic free vibration solutions under these more complex BCs. The nanocomposites’ material properties are first computed according to the rule of mixture. The Hamiltonian-system governing equation for the free vibration of doubly-curved panels is then formulated from the Donnell-Mushtari theory, and is solved by adopting the analytic symplectic superposition method. The obtained analytic solutions are derived without requiring predefined solution forms, and have been thoroughly validated by comparison with the results from the finite element method. By utilizing the accurate analytic solutions, the effects of aspect ratios, BCs, types of CNT distributions, and volume fractions of CNT on the free vibration behaviors are further analyzed. The present solution procedure and the resulting analytic solutions are expected to be useful for dynamic modeling of composite shell panels, supporting both future research and practical applications.

Peran Orang Tua Dalam Mendukung Pembelajaran Matematika di Rumah Bagi Siswa Sekolah Dasar

This study aims to understand the critical role of parents in supporting children's mathematics learning at the elementary school level. In this context, the research employs a qualitative approach with a case study design to explore parental involvement in facilitating mathematics comprehension at home and collaboration with teachers. Data were collected through in-depth interviews, participatory observations, and document analysis of ten purposively selected families. The results indicate that effective communication between parents and teachers, as well as the support provided at home, plays a significant role in enhancing children's understanding and motivation in learning mathematics. Parents who are actively involved in their children's learning, both through direct interaction with teachers and by facilitating learning activities at home, are proven to help children overcome challenges in learning mathematics. The study also emphasizes the importance of consistent collaboration between home and school to create a supportive learning environment. The implications of this study suggest that active parental involvement and collaboration with teachers are crucial for achieving optimal mathematics learning outcomes.

On the pricing of vulnerable Parisian options

Parisian options, serving as substitutes for barrier options, are frequently embedded in complex financial derivatives. Despite the considerable mathematical challenges in pricing, their significant applications in finance and insurance have generated extensive studies in the literature. However, the impact of counterparty risk has not been taken into account so far. To address this gap, we develop a closed-form pricing framework for vulnerable Parisian options. Utilizing the Laplace transform and measure-change technique, we derive closed-form pricing formulas of Parisian options incorporating counterparty credit risk. Finally, we conduct numerical analyses to verify our pricing formulas’ accuracy and efficiency.

Diagnostic benchmarking of many-objective evolutionary algorithms for real-world problems

Despite progress in multiobjective evolutionary algorithms (MOEAs) research, their efficacy in real-world scenarios remains unclear. This article introduces a diagnostic benchmarking framework to evaluate MOEAs, comprising (1) flexible MOEA construction software, (2) performance evaluation metrics and (3) real-world applications for benchmarking, reflecting diverse mathematical challenges. Utilizing this framework, NSGA-II, NSGA-III, RVEA, MOEA/D and Borg MOEA were evaluated across four applications with three to ten objectives. Collectively, the four applications capture challenges such as stochastic objectives, severe constraints, nonlinearity and complex Pareto frontiers. The study demonstrates how MOEAs that have shown strong performance on standard test problems can struggle on real-world applications. The benchmarking framework and results have value for enhancing the design and use of MOEAs in real-world applications. Further, the results highlight the need to improve the adaptability and ease-of-use of MOEAs given the often ill-defined nature of real-world problem-solving.

Quantum geometrodynamics revived I. Classical constraint algebra

Abstract In this series of papers, we present a set of methods to revive quantum geometrodynamics which encountered numerous mathematical and conceptual challenges in its original form promoted by Wheeler and De Witt. In this paper, we introduce the regularization scheme on which we base the subsequent quantization and continuum limit of the theory. Specifically, we employ the set of piecewise constant fields as the phase space of classical geometrodynamics, resulting in a theory with finitely many degrees of freedom of the spatial metric field. As this representation effectively corresponds to a lattice theory, we can utilize well–known techniques to depict the constraints and their algebra on the lattice. We are able to compute the lattice corrections to the constraint algebra. This model can now be quantized using the usual methods of finite–dimensional quantum mechanics, as we demonstrate in the following paper. The application of the continuum limit is the subject of a future publication.

Advancements in Q‐learning meta‐heuristic optimization algorithms: A survey

AbstractThis paper reviews the integration of Q‐learning with meta‐heuristic algorithms (QLMA) over the last 20 years, highlighting its success in solving complex optimization problems. We focus on key aspects of QLMA, including parameter adaptation, operator selection, and balancing global exploration with local exploitation. QLMA has become a leading solution in industries like energy, power systems, and engineering, addressing a range of mathematical challenges. Looking forward, we suggest further exploration of meta‐heuristic integration, transfer learning strategies, and techniques to reduce state space.This article is categorized under: Technologies > Computational Intelligence Technologies > Artificial Intelligence

Adaptive deep neural network optimized control for a class of nonlinear strict‐feedback systems with prescribed performance

SummaryIn this article, an adaptive deep neural network (DNN) optimized control strategy is developed for a class of nonlinear strict‐feedback systems with prescribed performance. First, the DNN is applied to approximate the unknown function, and the weight update law is designed to reduce the mathematical challenge based on the first‐order Taylor's series. Second, the optimized backstepping technique is utilized to construct virtual and actual controllers in the backstepping process to achieve the overall control optimization of the system. Next, a control strategy based on the time‐varying switching function and the quartic barrier Lyapunov function is employed to achieve the prescribed performance. Then, the tracking error can converge to the prescribed accuracy within the prescribed time, and every signal within the system has a bound. Finally, the particle swarm optimization algorithm is utilized to search for the designed parameters and simulation examples to verify the effectiveness of the control strategy.

Rheological behavior of the synovial fluid: a mathematical challenge

BackgroundSynovial fluid (SF) is often used for diagnostic and research purposes as it reflects the local inflammatory environment. Owing to its complex composition, especially the presence of hyaluronic acid, SF is usually viscous and non-homogeneous. The presence of high-molar-mass hyaluronan in this fluid gives it the required viscosity for its function as a lubricant. Viscosity is the greatest major hydraulic attribute of the SF in articular cartilage.MethodsEmpirical modeling of previously published results was performed. In this study, we explored the flow of a non-Newtonian fluid that could be used to model the SF flow. Analyzing the flow in a simple geometry can help explain the model’s efficacy and assess the SF models. By employing some viscosity data reported elsewhere, we summarized the dynamic viscosity values of normal human SF of the knee joints in terms of time after injecting hyaluronidase (HYAL) at 25°C. The suggested quadratic behavior was obtained through extrapolation. For accurate diagnosis or prediction, the comparison between three specific parameters (ai, t0, and ln η0) was made for normal and pathological cases under the same experimental conditions for treatment by addition of HYAL and for investigation of the rheological properties. A new model on the variation of viscosity on the SF of knee joints with time after injection of HYAL with respect to normal and postmortem samples at different velocity gradients was proposed using data previously reported elsewhere.ResultsThe rheological behavior of SF changes progressively over time from non-Newtonian to a Newtonian profile, where the viscosity has a limiting constant value (η0) independent of the gradient velocity at a unique characteristic time (t0 ≈ 8.5 h). The proposed three-parameter model with physical meaning offers insights into future pathological cases. The outcomes of this work are expected to offer new perspectives for diagnosis, criteria, and prediction of pathological case types through comparisons with new parameter values treated under the same experimental conditions as HYAL injection. This study also highlights the importance of HYAL treatment for better intra-assay precision.

Analytic Modelling of 2-D Transient Heat Conduction with Heat Source Under Mixed Boundary Constraints by Symplectic Superposition

Abstract Many studies have been conducted on 2-D transient heat conduction, but analytic modelling is still uncommon for the cases with complex boundary constraints due to the mathematical challenge. With an unusual symplectic superposition method, this paper reports new analytic solutions to 2-D isotropic transient heat conduction problems with heat source over a rectangular region under mixed boundary constraints at an edge. With the Laplace transform, the Hamiltonian governing equation is derived. The applicable mathematical treatments, e.g., the variable separation and the symplectic eigenvector expansion in the symplectic space, are implemented for the fundamental solutions whose superposition yields the ultimate solutions. Benchmark results obtained by the present method are tabulated, with verification by the finite element solutions. Instead of the conventional Euclidean space, the present symplectic-space solution framework has the superiority on rigorous derivations without pre-determining solution forms, which may be extended to more issues with the complexity caused by mixed boundary constraints.

Noise Enhances Excitability of a Neuronal Population with Heterogeneous Excitatory Neurons

AbstractCortical excitability is sensitive to noise perturbations despite homeostatic regulation. However, modeling the effect of noise in a heterogeneous neuronal population is a mathematical challenge. Using a mean‐field model of a randomly connected excitatory population, we investigated how noise affects excitability at the resting state. Stable equilibrium analysis revealed that increasing noise intensity enhances excitability, indexed by synaptic conductance. Further, this enhancement disappeared when excitability is sufficiently high due to strong mean input current, but not influenced by the connection probability of the population. Finally, the mechanism of noise‐induced enhancement was explained by eigenvalues moving toward the imaginary axis. © 2024 Institute of Electrical Engineers of Japan and Wiley Periodicals LLC.

Ephemeris accuracy improvement for moons of gas giants: a deep learning based method

Ephemerides accuracy of gas giant planetary system is of paramount importance in astronomical research, planetary exploration missions, and space navigation. Traditional mathematical methods face challenges when processing with data of different observation precisions, which may lead to extra noise and reduce the ephemeris accuracy. This paper proposes an Denoising Autoencoder based Method for Ephemeris Improvement (DAMEI) to improve the accuracy of ephemerides for moons in gas giant planetary system. Utilizing multiple sources of data efficiently, the DAMEI method can mitigate the impact of noise and uncertainty introduced by observational data with different precisions. Based on the symmetrical structure, the DAMEI method learns to encode essential motion features of gas giant planetary system into a latent space and captures the intricate patterns in planetary motion from observational data, subsequently decodes it to improve the ephemeris accuracy. The experimental results show that, for Jupiter’s major satellites (Galilean moons), the DAMEI method achieves more accurate ephemeris of up to 91.65% compared with current mathematical method. The proposed method is also assessed with satellites of Saturn, Uranus, and Neptune. It is shown that the DAMEI method also presents a better performance of up to 95.37%. The promising performance of DAMEI method can reduce the uncertainty introduced by low-accuracy data and improve ephemeris accuracy making the utmost of observational data with different precisions.