Mathematical Complexity Research Articles

Naturally circulated system under low to moderate heating condition with supercritical fluid: A comprehensive investigation of loop orientation and Ledinegg instability

Out of the two stability characteristics i.e. static and dynamics, static instability has hardly been explored in such systems due to mathematical complexity and absence of explicit boundary conditions. Present numerical investigation deals with the 1D numerical framework analyzing the steady-state and static instability of the loop under different loop orientation i.e., horizontal heater horizontal cooler (HHHC), vertical heater horizontal cooler (VHHC), horizontal heater vertical cooler (HHVC) and vertical heater vertical cooler (VHVC). The range of heating power considered is applicable in the low to moderate heating condition such as in solar heater and electronic chip cooling. Due to uneven generation of buoyancy and friction throughout the flow path, occurrence of FiHTD is also dependent on it, with best and worst thermalhydraulic characteristics for HHHC and VHVC respectively. For every condition, the appearance of static instability was found to be within the heating power limit. For T∞ = 295 K it spanned over 1050 W for HHHC orientated loop which got decreased by 9.5%, 23.8% and 65.7% for VHHC, HHVC and VHVC respectively. Under the condition of increased of T∞ = 300 K the same was found within the limit of 575 W and got decreased by 13%, 21.7% and 64.3% for cases mentioned above insequential manner. Stability maps drawn using well known non-dimensional numbers holds a good qualitative behavior of static instability. Dynamic instability is well explored for HHHC combination has throughly done previously by various authors and hence, has not been considered here. The authors believe that dynamic instability will not be much more prominant due to predefined flow direction and can be explored as a future work.

Enhancing stability and flexibility of platoon control with an adaptive spacing policy

Connected and Automated Vehicles (CAVs) forming platoons have shown significant potential in enhancing traffic flow. This paper proposes a cooperative platoon control model using MPC integrated with the CTG spacing policy to improve both platoon stability and overall traffic performance. This integration offers greater flexibility and practicality compared to the CS-based model. Additionally, the control model includes a safety constraint with high flexibility to prevent collisions. We numerically analyse the asymptotic stability of the proposed control model to address its mathematical complexity. A wide range of parameter configurations and weight assignment schemes were tested to provide rigorous proof of the model's asymptotic stability. Numerical experiments show that the CTG-based model offers better stability than the CS-based model, though it leads to larger spacings at higher vehicle speeds. Moreover, traffic simulation results demonstrate that the CTG-based model outperforms the CS-based model in mitigating traffic oscillations and reducing congestion.

Unlocking AutoML: Enhancing Data with Deep Learning Algorithms for Medical Imaging

Deep learning algorithms have become increasingly popular over the years, having proved their efficiency in input-output functions for distinct types of data. This technology is particularly useful in medical imaging, where complex image structures often generate disagreements between medical staff. These technologies can streamline the diagnostic process by performing automatic image analysis, which results in more accurate and reproducible diagnoses. Additionally, these technologies can enhance content retrieval systems by automatically labeling the images based on the structures they possess. Despite the benefits, the mathematical complexity of deep learning algorithms and their training optimizations can be challenging. Automated machine learning provides a solution to this challenge by offering tools that automate the development and training of these algorithms. This makes it possible for users with limited programming experience to take advantage of these powerful technologies to quickly develop and prototype analysis algorithms for their specific needs. This paper presents a management platform for deep learning services on the cloud that provides a code-free experience through automated machine learning. The evaluation was done in one of the most demanding scenarios, where the service was integrated into a research pathology PACS to annotate mitotic cells in breast cancer tissue automatically. The annotations are processed by an open-source PACS archive and stored directly on the files, enhancing the image metadata and consequently content retrieval systems. The results of the developed algorithms were compared to the state-of-the-art to evaluate the competitiveness of the solution.

Analytical approach in designing PID controller for complex fractional order transfer function

The study focuses on the fractional complex order plant model, which has gained popularity in applied mathematics, physics, and control systems. A significant contribution of this research lies in discussing the physical phenomena associated with complex plant models and their impact on system stability and robustness. The main purpose of the method presented in this paper is to tune the controller parameters to ensure the stability and robustness of the system. There are methods presented in the literature for this purpose. One of these methods is to keep the phase curve in the system frequency response flat within a certain range. However, this process is based on equating the derivative of the phase value to zero at a certain frequency and adds great mathematical complexity to the calculations. In this study, reliable analytical formulas are presented for the same purpose using a graphical approach. Since the fractional complex order plant model represents the most general mathematical form, it enables easy creation of other plant models, including integer order and fractional order plant. The reason why this plant is chosen is that this structure can be named as the universal plant, which all other structures can be built by making little variations. For instance, a transfer function having integer, real and/or complex number coefficients and/or orders can be obtained by proper determination of the parameters of the universal plant. A time delay can also be added towards researcher's desire. The main inspiration comes from studying on an inclusive plant. The method in this paper intends to tune the well‐known classical Proportional Integral Derivative controller. Thus, effectiveness of the integer order controller on various plants will be shown. This approach provides analytical calculation equations for the physical modifications of plants with integer, fractional, and/or complex coefficients and/or orders. The effectiveness of the method is demonstrated visually with different examples that include these different possible situations. The results observed in the changes of the parameters in the transfer functions were also examined. Thus, pros and cons of the variations of integer, fractional, and complex numbers on system parameters have been shown.

Computational toolbox for scattering of focused light from flattened or elongated particles using spheroidal wavefunctions

T-matrix methods, with incident and scattered fields described as sums of multipolar fields, are attractive computational methods for many scattering problems due to their versatility, accuracy, and computational efficiency, especially for repeated calculations. However, numerical difficulties often hamper their use for non-spherical particles with large aspect ratios. Further, even if far-field scattering can be accurately calculated, it can be impossible to accurately calculate near-fields. The use of spheroidal wavefunctions, instead of the commonly-used spherical wavefunctions, can be a useful solution for these problems. However, the mathematical complexity of spheroidal wavefunctions, and the challenging task of accurate numerical calculation of them, are significant barriers to their use. We have developed a computational package of MATLAB routines for performing electromagnetic scattering calculations using spheroidal wavefunctions. These allow the determination of light scattering by non-spherical particles with high aspect ratios that would be inaccessible for double precision calculation using spherical wavefunctions. We demonstrate that our codes can be successfully used for cylinders of aspect ratios from 1/10 and 10, and for metal nanoparticles. We include routines for interoperability with regular T-matrix codes and packages such as our optical tweezers toolbox.

"In Mathematical Language": On Mathematical Foundations of Quantum Foundations.

The argument of this article is threefold. First, the article argues that from its rise in the sixteenth century to our own time, the advancement of modern physics as mathematical-experimental science has been defined by the invention of new mathematical structures. Second, the article argues that quantum theory, especially following quantum mechanics, gives this thesis a radically new meaning by virtue of the following two features: on the one hand, quantum phenomena are defined as essentially different from those found in all previous physics by purely physical features; and on the other, quantum mechanics and quantum field theory are defined by purely mathematical postulates, which connect them to quantum phenomena strictly in terms of probabilities, without, as in all previous physics, representing or otherwise relating to how these phenomena physically come about. While these two features may appear discordant, if not inconsistent, I argue that they are in accord with each other, at least in certain interpretations (including the one adopted here), designated as "reality without realism", RWR, interpretations. This argument also allows this article to offer a new perspective on a thorny problem of the relationships between continuity and discontinuity in quantum physics. In particular, rather than being concerned only with the discreteness and continuity of quantum objects or phenomena, quantum mechanics and quantum field theory relate their continuous mathematics to the irreducibly discrete quantum phenomena in terms of probabilistic predictions while, at least in RWR interpretations, precluding a representation or even conception of how these phenomena come about. This subject is rarely, if ever, discussed apart from previous work by the present author. It is, however, given a new dimension in this article which introduces, as one of its main contributions, a new principle: the mathematical complexity principle.

Accessing and Assessing Components of Elementary and Middle School Students’ Mathematical Disposition Through Metaphors

ABSTRACT This study examined student-generated metaphors comparing food to math as a means of accessing and assessing components of prekindergarten through Grade 8 students’ (N = 306) mathematical disposition. Previous research provided insight into predominantly affective components of older students’ mathematical disposition (i.e. Grade 4 through college). However, we extend these findings to include nonaffective components and younger students. Our study highlights the value in conceptualizing mathematical disposition as the sum of three mental functions, namely cognitive (e.g., mental processes such as reasoning), affective (e.g., attitudes, feelings, beliefs about mathematics or oneself as a learner), and conative (e.g., effort, grit, or level of challenge), and whose components may span multiple categories of these mental functions. Our study also extends the research on mathematical disposition by revealing that it is conditional in addition to being complex. Furthermore, several notable grade-based trends from data emerged in relation to students’ enjoyment of mathematics as well as their views regarding the applicability, prevalence, and variety and complexity of mathematics.

Exact Expressions for Kullback-Leibler Divergence for Univariate Distributions.

The Kullback-Leibler divergence (KL divergence) is a statistical measure that quantifies the difference between two probability distributions. Specifically, it assesses the amount of information that is lost when one distribution is used to approximate another. This concept is crucial in various fields, including information theory, statistics, and machine learning, as it helps in understanding how well a model represents the underlying data. In a recent study by Nawa and Nadarajah, a comprehensive collection of exact expressions for the Kullback-Leibler divergence was derived for both multivariate and matrix-variate distributions. This work is significant as it expands on our existing knowledge of KL divergence by providing precise formulations for over sixty univariate distributions. The authors also ensured the accuracy of these expressions through numerical checks, which adds a layer of validation to their findings. The derived expressions incorporate various special functions, highlighting the mathematical complexity and richness of the topic. This research contributes to a deeper understanding of KL divergence and its applications in statistical analysis and modeling.

A blockchain-assisted privacy-preserving signature scheme using quantum teleportation for metaverse environment in Web 3.0

At the forefront of next-generation internet technology, the concept of the metaverse is gaining traction. It unifies the virtual and physical worlds into a single virtual realm and has the potential to revolutionize social networks, gaming, healthcare and education. Quantum teleportation, renowned for its ability to ensure secure and reliable communications, is set to transform interactions within this immersive digital realm. Nevertheless, the conventional cryptosystems that typically rely on mathematical complexity will no longer be dependable and secure with the advancement of quantum computing. Applied to blockchain architecture, existing post-quantum cryptography technologies, which raise the time complexity of the algorithm, will result in relatively low efficiency and substantial resource overhead. In contrast, quantum cryptography has the ability to improve blockchain’s security and efficiency. Moreover, the reliance on centralized authorities in current metaverse platforms highlights the need for decentralization to improve interoperability and security. Web 3.0 technologies offer a solution by enabling a decentralized metaverse ecosystem. Therefore, the proposed approach leverages quantum teleportation to achieve secure communication and the integration of quantum cryptography and Web 3.0 enhances the efficiency, security and realism of metaverse environments. Through rigorous safety and efficiency analysis, we demonstrate the protocol’s robustness and performance, ensuring adherence to fundamental security properties such as unforgeability, undeniability, verifiability, and traceability.

MDTA: An efficient, scalable and fast Multiple Disjoint Tree Algorithm for dynamic environments

Emerging applications such as telemedicine, the tactile Internet or live streaming place high demands on low latency to ensure a satisfactory Quality of Experience (QoE). In these scenarios the use of trees can be particularly interesting to efficiently deliver traffic to groups of users because they further enhance network performance by providing redundancy and fault tolerance, ensuring service continuity when network failure or congestion scenarios occur. Furthermore, if trees are isolated from each other (they do not share common communication elements as links and/or nodes), their benefits are further enhanced since events such as failures or congestion in one tree do not affect others. However, the challenge of computing fully disjoint trees (both link- and node-disjoint) introduces significant mathematical complexity, resulting in longer computation times, which negatively impacts latency-sensitive applications.In this article, we propose a novel algorithm designed to rapidly compute multiple fully (either link- or node-) disjoint trees while maintaining efficiency and scalability, specifically focused on targeting the low-latency requirements of emerging services and applications. The proposed algorithm addresses the complexity of ensuring disjointedness between trees without sacrificing performance. Our solution has been tested in a variety of network environments, including both wired and wireless scenarios.The results showcase that our proposed method is approximately 100 times faster than existing techniques, while achieving a comparable success rate in terms of number of obtained disjoint trees. This significant improvement in computational speed makes our approach highly suitable for the low-latency requirements of next-generation networks.

Improving the CSIDH Protocol for Multi-party Cryptography: Rigorous Mathematical Analysis, Efficiency, and Security Comparison

This paper introduces a novel Distributed Key Generation (DKG) protocol based on the Commutative Supersingular Isogeny Diffie-Hellman (CSIDH) framework for secure multi-party cryptography. Our proposed protocol is designed to address scalability and security concerns, particularly in post-quantum cryptographic systems. The main contributions include the introduction of Piecewise Verifiable Proofs (PVPs) for non-interactive zero-knowledge verification of secret shares, and the provision of rigorous security analysis, including resistance to quantum adversaries via Shor’s and Grover’s algorithms. We analyze the protocol’s efficiency, ensuring low computational overhead even in large-scale systems, and compare it with other distributed cryptographic protocols such as RSA-based and lattice-based schemes. Through mathematical proofs and complexity analysis, we demonstrate that our protocol offers enhanced security, efficiency, and scalability in a post-quantum environment. The results presented in this paper provide a strong foundation for implementing secure multi-party computations in quantum-resistant systems.

HARNESSING GROUP THEORY FOR SIMPLIFIED ELECTRIC CIRCUIT ANALYSIS

Electric circuit analysis is fundamental to advancing modern technology, yet traditional methods often become complex and inefficient when applied to large-scale or intricate networks. Group theory, a branch of abstract algebra renowned for its capacity to exploit symmetry, has shown promise as a solution to this challenge. By applying group theoretical principles to circuit analysis, particularly through symmetry operations and transformation groups, engineers can simplify complex systems, reducing computational demands and revealing deeper insights into circuit behavior. Through systematic symmetry-based simplifications, group theory can decrease the number of equations required by up to 40% in certain configurations, enhancing both analytical efficiency and practical implementation. However, challenges include the mathematical complexity inherent to group theory, the limitations in analyzing highly irregular or nonlinear circuits, and the need for compatible computational tools. Recommendations include the integration of group theoretical approaches into standard circuit analysis software, focused training for electrical engineers, and further research to address the application of these methods in emerging technologies such as quantum and neuromorphic computing. Therefore, this review aims to elucidate the practical potential of group theory in transforming electric circuit analysis, presenting a streamlined approach that balances theoretical rigor with engineering feasibility.

Constitutive modelling in computational geomechanics – 61st Rankine lecture, British Geotechnical Association, 2023

Constitutive models are an essential part of computational modelling in geotechnics; they are at the heart of almost all theoretical predictions of geotechnical structures. How the stress–strain (and perhaps time) response of soil (and rock) is represented in these mathematical models is usually the key to successful prediction of the behaviour of geotechnical structures. However, the important details of these models, particularly the idealisations that are made, may be poorly or incompletely understood, or ignored, sometimes at significant cost to the unwary analyst. Indeed, the capabilities and the shortcomings of these models, especially the more advanced models, are not always easy to ascertain. In some cases, determination of the values of the input parameters is not straightforward. Consequently, it may be difficult to determine which model to select for a particular task. This paper explores some of the more important developments in the constitutive modelling of soils and addresses some of these issues of potential concern. The need for such models and the various attributes and capabilities that the commonly used models possess are reviewed. Also discussed is the issue of matching a particular model to the geotechnical problem at hand, which model attributes are required and why. The intention is to place emphasis on the physical basis of these models, rather than explore their mathematical complexity in detail. Some of the constitutive models encoded in the software packages used routinely in geotechnical practice are reviewed, and discussion is also provided on their specific limitations. Examples of practical applications, involving the solution of boundary and initial value problems, are described to illustrate both the advantages and some of the limitations of both commonly used and highly advanced constitutive models.

Designing a resilient hydrogen hub under disruption risks and non-stationary demand distribution

Hydrogen is a viable and sustainable energy alternative. It offers a solution to mitigate greenhouse gases and fortify energy security. However, its supply chain faces many uncertainties and challenges, including demand fluctuations and sourcing disruptions. This paper introduces an innovative two-stage stochastic model to plan hydrogen hub procurement, storage, and sales. In the first stage, the model optimises the order quantities by considering real-time inventory levels. This forward-thinking strategy aims to improve operational efficiency and adaptability. In the second stage, the model refines the hub operations by incorporating supplier resilience, exploration of alternative markets, emission considerations from each source, and terminal connection planning. Integrating these elements contributes to a comprehensive framework for robust hydrogen hub scheduling. This paper adopts a Benders decomposition algorithm to address the mathematical complexity of the model. This approach is necessary to ensure a smooth and efficient computational process. Empirical testing and validation of the developed model, along with the robustness of the solution methodology, emphasise its effectiveness in handling uncertainties and disruptions. This paper contributes to the existing literature by shedding light on critical facets of disruption management, supplier resilience, and emissions reduction within the hydrogen supply chain design.

AI-Lorenz: A physics-data-driven framework for Black-Box and Gray-Box identification of chaotic systems with symbolic regression

Discovering mathematical models that characterize the observed behavior of dynamical systems remains a major challenge, especially for systems in a chaotic regime, due to their sensitive dependence on initial conditions and the complex non-linear interactions within the system. The challenge is even greater when the physics underlying such systems is not yet understood, and scientific inquiry must solely rely on empirical data. Despite advancements such as the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm, which has shown success in identifying chaotic systems with reasonable accuracy, challenges remain in dealing with noise, sparse data, and the need for models that generalize well across different chaotic systems. Driven by the need to fill this gap, we develop a framework named AI-Lorenz that learns mathematical expressions modeling complex dynamical behaviors by identifying differential equations from noisy and sparse observable data. We train a physics-informed neural network (PINN) with the eXtreme Theory of Functional Connections (X-TFC) algorithm by using data and known physics (when available) to learn the dynamics of a system, its rate of change in time, and missing model terms, which are used as input for a symbolic regression algorithm (PySR) to autonomously distill the explicit mathematical terms. This, in turn, enables us to predict the future evolution of the dynamical behavior. The performance of this framework is validated by recovering the right-hand sides and unknown terms of certain complex, chaotic systems, such as the well-known Lorenz system, a six-dimensional hyperchaotic system, the non-autonomous Sprott chaotic system, and the slow-fast Duffing system, and comparing them with their known analytical expressions and state-of-the-art regression and system identification methods.

Minimum Risk Quantification Method for Error Threshold of Wind Farm Equivalent Model Based on Bayes Discriminant Criterion

The error threshold is the cornerstone to balance the mathematical complexity and simulation speed of wind farm (WF) equivalent models, and can promote the standardization process of equivalent methodology. Due to differences in power system conditions and model evaluation standards in different countries, the form and indexes of error thresholds of WF equivalent models have not been unified yet. This paper proposes a theoretical method for quantifying the minimum risk of error threshold of WF equivalent models based on the Bayes discriminant criterion. Firstly, the Euclidean errors of WF equivalent models in different periods are quantified, and the probability density distributions of the errors are fitted by kernel density estimation. Secondly, the real-time weighted prior probability algorithm is used to obtain the prior probability of a valid WF equivalent model, and the different losses caused by the missed judgment and misjudgment of the model validity to power systems are taken into account. Thirdly, the minimum risk quantification model of error threshold is established based on the Bayes discriminant criterion, and the feasibility of the proposed method is verified by an actual WF with numerical examples. Compared with the existing error thresholds, the proposed error threshold can more quickly and accurately determine the validity of WF equivalent models.

Simple or complex statistical models: Non-traditional regression models with intuitive interpretations

The evolution of statistical modelling has historically been constrained by the practical limitations of computation; early statistical modelling favoured models which could feasibly be estimated. As increased mathematical complexity often implies more intricate computation, statistical models have grown both mathematically and computationally more complex. However, paradoxically, sometimes conceptually simpler models present more computational challenges than complex ones, and these have historically been neglected. In the case of binary responses, logistic regression models are the gold standard; covariates are modelled additively on the log-odds scale. In the case of time-to-event responses, the Cox proportional hazards regression model has additivity on the log-hazard rate scale. Both of these methods are computationally convenient, and yet the scales on which covariates are modelled are far from intuitive. We demonstrate the use of alternative models which are computationally more complex, yet feasible, but in which modelling is on a more interpretable scale. In the case of binary responses, a more intuitive alternative is log-binomial regression, in which modelling is on the log-relative risk scale. In the case of time-to-event responses, distributional regression enables modelling on the time scale. While logistic regression and proportional hazards regression qualitatively both deliver the same conclusions as the alternative models, log-binomial and distributional regression provide more interpretable coefficients, which are readily estimated.

A Generalized Method for Deriving Steady-State Behavior of Consistent Fuzzy Priority for Interdependent Criteria

Interdependent criteria play a crucial role in complex decision-making across various domains. Traditional methods often struggle to evaluate and prioritize criteria with intricate dependencies. This paper introduces a generalized method integrating the analytic network process (ANP), the decision-making trial and evaluation laboratory (DEMATEL), and the consistent fuzzy analytic hierarchy process (CFAHP) in a fuzzy environment. The Drazin inverse technique is applied to derive a fuzzy total priority matrix, and we normalize the row sum to achieve the steady-state fuzzy priorities. A numerical example in the information systems (IS) industry demonstrates the approach’s real-world applications. The proposed method derives narrower fuzzy spreads compared to the past fuzzy analytic network process (FANP) approaches, minimizing objective uncertainty. Total priority interdependent maps provide insights into complex technical and usability criteria relationships. Comparative analysis highlights innovations, including non-iterative convergence of the total priority matrix and the ability to understand interdependencies between criteria. The integration of the FANP’s network structure with the fuzzy DEMATEL’s influence analysis transcends the capabilities of either method in isolation, marking a significant methodological advancement. By addressing challenges such as parameter selection and mathematical complexity, this research offers new perspectives for future research and application in multi-attribute decision-making (MADM).

Infinite-Horizon Probability of Ruin for a Variable-Memory Counting Process (Hawkes Process)

The reserve R(t) of an insurance company denotes the accumulated capital up to time ttt throughout its operational span. While this reserve can be modeled using stochastic processes, the endeavor remains notably intricate. The mathematical complexity presents a considerable obstacle for researchers within this domain. Prior research has predominantly focused on reserve models constructed from Markovian processes. In this manuscript, our attention shifts to risk models derived from non-Markovian processes, particularly those with claim arrivals governed by Hawkes processes. Before delving into the core subject matter, we provide an extensive review of the literature on counting processes with variable memory. This includes a thorough exploration of Hawkes processes, Gerber-Shiu functions, integro-differential equations, and Laplace transforms. These elements are crucial for computing the probability of ruin over an infinite time horizon, especially in scenarios where interdependence exists between inter-claim times. Additionally, we present the requisite mathematical tools essential for comprehending the contents of this article, specifically tailored for individuals with a foundational understanding of actuarial science. Furthermore, we have successfully determined the probability of ruin, marking a significant milestone in our investigation.

Motion of a cylindrical net with a mass at the end

The motion of a semi-infinite cylindrical net with a mass at one end is examined. The cylindrical net, with a mass at one end, is initially stretched. Addressing the problem of wave propagation in deformable filament systems, accounting for significant deviations from their original rectilinear shape, presents considerable mathematical complexity, as the equations of motion constitute a system of nonlinear partial differential equations. The solution, derived using characteristic equations, reveals the occurrence of traveling waves. This solution is constructed by satisfying conditions at the point of contact between the net and the load. The results are presented in the form of graphs and tables.