AppliedMath, Volume 4, Issue 3 (September 2024) – 21 articles

The two-parameter logistic (2PL) item response theory (IRT) model is frequently applied to analyze group differences for multivariate binary random variables. The item parameters in the 2PL model are frequently fixed when estimating the mean and the standard deviation for a group of interest. This method is also called fixed item parameter calibration (FIPC). In this article, the bias and the linking error of the FIPC approach are analytically derived in the presence of random uniform differential item functioning (DIF). The adequacy of the analytical findings was validated in a simulation study. It turned out that the extent of the bias and the variance in distribution parameters increases with increasing variance of random DIF effects. Full article

This paper introduces the Accumulated Concept Graph (ACG), a visualization tool based on the quantile method designed to analyze three-way data, including distributional data. Such data often have complex structures that make it difficult to identify patterns using conventional visualization techniques. The ACG represents each object with one or more monotonic line graphs. As a result, the three-way data are visualized as a set of parallel monotonic line graphs that never intersect. This non-intersecting property allows for the easy identification of both macroscopic and microscopic patterns within the data. We demonstrate the usefulness of ACGs and principal component analysis in the analysis of real three-way datasets. Full article

In this paper, an innovative hybrid technique is proposed for the efficient training of artificial neural networks, which are used both in class learning problems and in data fitting problems. This hybrid technique combines the well-tested technique of Genetic Algorithms with an innovative variant of Simulated Annealing, in order to achieve high learning rates for the neural networks. This variant was applied periodically to randomly selected chromosomes from the population of the Genetic Algorithm in order to reduce the training error associated with these chromosomes. The proposed method was tested on a wide series of classification and data fitting problems from the relevant literature and the results were compared against other methods. The comparison with other neural network training techniques as well as the statistical comparison revealed that the proposed method is significantly superior, as it managed to significantly reduce the neural network training error in the majority of the used datasets. Full article

The classical tests for combining p-values use suitable statistics $T(P_1,\dots ,P_n)$, which are based on the assumption that the observed p-values are genuine, i.e., under null hypotheses, are observations from independent and identically distributed $\mathrm{Uniform}(0,1)$ random variables $P_1,\dots ,P_n$. However, the phenomenon known as publication bias, which generally results from the publication of studies that reject null hypotheses of no effect or no difference, can tempt researchers to replicate their experiments, generally no more than once, with the aim of obtaining “better” p-values and reporting the smallest of the two observed p-values, to increase the chances of their work being published. However, when such “fake p-values” exist, they tamper with the statistic $T(P_1,\dots ,P_n)$ because they are observations from a $\mathrm{Beta}(1,2)$ distribution. If present, the right model for the random variables $P_k$ is described as a tilted Uniform distribution, also called a Mendel distribution, since it was underlying Fisher’s critique of Mendel’s work. Therefore, methods for combining genuine p-values are reviewed, and it is shown how quantiles of classical combining test statistics, allowing a small number of fake p-values, can be used to make an informed decision when jointly combining fake (from Two P) and genuine (from not Two P) p-values. Full article

Magnetic resonance imaging (MRI) is crucial for its superior soft tissue contrast and high spatial resolution. Integrating deep learning algorithms into MRI reconstruction has significantly enhanced image quality and efficiency. This paper provides a comprehensive review of optimization-based deep learning models for MRI reconstruction, focusing on recent advancements in gradient descent algorithms, proximal gradient descent algorithms, ADMM, PDHG, and diffusion models combined with gradient descent. We highlight the development and effectiveness of learnable optimization algorithms (LOAs) in improving model interpretability and performance. Our findings demonstrate substantial improvements in MRI reconstruction in handling undersampled data, which directly contribute to reducing scan times and enhancing diagnostic accuracy. The review offers valuable insights and resources for researchers and practitioners aiming to advance medical imaging using state-of-the-art deep learning techniques. Full article

This article provides an exact formula for the survival probability of Brownian motion with drift when the absorbing boundary is defined as an intermittent step barrier, i.e., an alternate sequence of time intervals when the boundary is piecewise constant, and time intervals without any defined boundary. Numerical implementation is dealt with by a simple and robust Monte Carlo integration algorithm directly derived from the formula, which compares favorably with conditional Monte Carlo simulation. Exact analytical benchmarks are also provided to assess the accuracy of the numerical implementation. Full article

Cancer, a complex disease characterized by uncontrolled cell growth and metastasis, remains a formidable challenge to global health. Mathematical modeling has emerged as a critical tool to elucidate the underlying biological mechanisms driving tumor initiation, progression, and treatment responses. By integrating principles from biology, physics, and mathematics, mathematical oncology provides a quantitative framework for understanding tumor growth dynamics, microenvironmental interactions, and the evolution of cancer cells. This study explores the key applications of mathematical modeling in oncology, encompassing tumor growth kinetics, intra-tumor heterogeneity, personalized medicine, clinical trial optimization, and cancer immunology. Through the development and application of computational models, researchers aim to gain deeper insights into cancer biology, identify novel therapeutic targets, and optimize treatment strategies. Ultimately, mathematical oncology holds the promise of transforming cancer care by enabling more precise, personalized, and effective therapies. Full article

This work considers the existence of solutions of the heteroclinic type in nonlinear second-order differential equations with $\varphi$-Laplacians, incorporating generalized impulsive conditions on the real line. For the construction of the results, it was only imposed that $\varphi$ be a homeomorphism, using Schauder’s fixed-point theorem, coupled with concepts of $L^1$-Carathéodory sequences and functions along with impulsive points equiconvergence and equiconvergence at infinity. Finally, a practical part illustrates the main theorem and a possible application to bird population growth. Full article

This study investigates the time-series behavior of vegetable prices in the Central Market of Thessaloniki, Greece, using Recurrence Plot (RP) analysis and Recurrence Quantification Analysis (RQA), which considers non-linearities and does not necessitate stationarity of time series. The period of study was 1999–2016 for practical and research reasons. In the present work, we focus on vegetables available throughout the year, exploring the dynamics and interrelationships between their prices to avoid missing data. The study applies RP visual inspection classification, a clustering based on RQA parameters, and a classification based on the RQA analysis graphs with epochs for the first time. The aim of the paper was to investigate the grouping of products based on their price dynamical behavior. The results show that the formed groups present similarities related to their use as dishes and their way of cultivation, which apparently affect the price dynamics. The results offer insights into market behaviors, helping to inform better management strategies and policymaking and offer a possibility to predict variability of prices. This information can interest government policies in various directions, such as what products to develop for greater stability, identity for fluctuating prices, etc. In future work, a larger dataset including missing data could be included, as well as a machine-learning algorithm to classify the products based on the RQA with epochs graphs. Full article

This paper explores the mathematical dynamics of consumer spending during a financial crisis using opponent process theory (OPT). Traditionally applied in psychology, OPT explains how initial emotional responses are followed by counteracting reactions to restore equilibrium. This study models the short-term boost in consumer spending and subsequent economic adjustments. Utilizing differential equations to represent these processes, this paper provides insights into the interplay between immediate policy effects and longer-term economic consequences. We focus on the United States (US) response to the 2008 Global Financial Crisis in this study. Results show evidence of diminishing response from prolonged stimuli due to demand saturation, resource allocation inefficiencies, and agent adaptation. Monetary stimuli may inflate debt/prices, outweighing benefits, and structural issues persist despite stimuli. Confidence and expectations impact response because perceived ineffectiveness weakens impact over time. Thus, while stimuli can initially boost activity, their sustained impact demands careful consideration of economic dynamics and agents’ responses. Full article

Multiple harmonic-like numbers are studied using the generating function approach. A closed form is stated for binomial sums involving these numbers and two additional parameters. Several corollaries and examples are presented which are immediate consequences of the main result. Finally, combinatorial identities involving harmonic-like numbers and other prominent sequences like hyperharmonic numbers and odd harmonic numbers are offered. Full article

Mathematician and philosopher Charles Howard Hinton posited a plausible correlation between higher-dimensional spaces, also referred to as ‘hyperspaces’, and the allegorical concept articulated by the Ancient Greek philosopher Plato in his work, Republic, known as the ‘Cave.’ In Plato’s allegory, individuals find themselves situated in an underground ‘Cave’, constrained by chains on their legs and neck, perceiving shadows and sound reflections from the ‘real’ world cast on the ‘Cave’ wall as their immediate reality. Hinton extended the interpretation of these ‘shadows’ through the induction method, asserting that, akin to a 3D object casting a 2D shadow, the ‘shadow’ of a 4D hyper-object would exhibit one dimension less, manifesting as a 3D object. Building upon this conceptual framework, the authors posit a correlation between the perceived acoustic space of the bounded individuals within the ‘Cave’ and the characteristics of a 4D acoustic space, a proposition substantiated mathematically by scientific inquiry. Furthermore, the authors introduce an interactive art application developed as a methodical approach to exploring the hypothetical 4D acoustic space within Plato’s ‘Cave’, as perceived by the bounded individuals and someone liberated from his constraints navigating through the ‘Cave.’ Full article

Fractional differential operators are inherently non-local, so global methods, such as spectral methods, are well suited for handling these non-local operators. Long-time integration of differential models such as chaotic dynamical systems poses specific challenges and considerations that make multi-domain numerical methods advantageous when dealing with such problems. This study proposes a novel multi-domain pseudospectral method based on the first kind of Chebyshev polynomials and the Gauss–Lobatto quadrature for fractional initial value problems.The proposed technique involves partitioning the problem’s domain into non-overlapping sub-domains, calculating the fractional differential operator in each sub-domain as the sum of the ‘local’ and ‘memory’ parts and deriving the corresponding differentiation matrices to develop the numerical schemes. The linear stability analysis indicates that the numerical scheme is absolutely stable for certain values of arbitrary non-integer order and conditionally stable for others. Numerical examples, ranging from single linear equations to systems of non-linear equations, demonstrate that the multi-domain approach is more appropriate, efficient and accurate than the single-domain scheme, particularly for problems with long-term dynamics. Full article

Scholars of English Literature unanimously say that J.R.R. Tolkien influenced C.S. Lewis’s writings. For the first time, we have investigated this issue mathematically by using an original multi-dimensional analysis of linguistic parameters, based on surface deep language variables and linguistic channels. To set our investigation in the framework of English Literature, we have considered some novels written by earlier authors, such as C. Dickens, G. MacDonald and others. The deep language variables and the linguistic channels, discussed in the paper, are likely due to writers’ unconscious design and reveal connections between texts far beyond the writers’ awareness. In summary, the capacity of the extended short-term memory required to readers, the universal readability index of texts, the geometrical representation of texts and the fine tuning of linguistic channels within texts—all tools largely discussed in the paper—revealed strong connections between The Lord of the Rings (Tolkien), The Chronicles of Narnia, The Space Trilogy (Lewis) and novels by MacDonald, therefore agreeing with what the scholars of English Literature say. Full article

Most optimisation research focuses on relatively simple cases: one decision maker, one objective, and possibly a set of constraints. However, real-world optimisation problems often come with complications: they might be multi-objective, multi-agent, multi-stage or multi-level, and they might have uncertainty, partial knowledge or nonlinear objectives. Each has led to research areas with dedicated solution methods. However, when new hybrid problems are encountered, there is typically no solver available. We define a broad class of discrete optimisation problem called an influence program, and describe a lightweight algorithm based on multi-agent multi-objective reinforcement learning with sampling. We show that it can be used to solve problems from a wide range of literatures: constraint programming, Bayesian networks, stochastic programming, influence diagrams (standard, limited memory and multi-objective), and game theory (multi-level programming, Bayesian games and level-k reasoning). We expect it to be useful for the rapid prototyping of solution methods for new hybrid problems. Full article

Voltage source converters (VSCs) have emerged as the key components in modern power systems, facilitating efficient energy conversion and flexible power flow control. Understanding the fundamental circuit model of VSCs is essential for their accurate modeling and analysis in power system studies. A basic voltage source converter circuit model connected to an LC filter is essential because it lowers the harmonic distortions and enhances the overall power quality of the micro-grid. This guarantees a clean and steady power supply, which is necessary for the integration of multiple renewable energy sources and sensitive loads. A comprehensive methodology for developing a basic circuit model of VSCs, focusing on the key components and principals involved, is presented in this paper. The methodology includes the modeling of space vector pulse-width modulation (SVPWM) as well as the direct quadrature zero synchronous reference frame. Different design controls, including the design of current control loop in the S-domain, the design of the direct current (DC) bus voltage control loop in the S-domain, and the design of the alternating current (AC) voltage control loop in the S-domain, are explored to capture the dynamic behavior and control strategies of VSCs accurately. The proposed methodology provides a systematic framework for modeling VSCs, enabling engineers and researchers to analyze their performance and assess their impact on power system stability and operation. Future studies can be conducted by using case studies and simulation scenarios to show the efficiency and applicability of the developed models in analyzing VSC-based power electronics applications, including high-voltage direct current (HVDC) transmission systems and flexible alternating current transmission systems (FACTS). The significance of this work lies in its potential to advance the understanding and application of VSCs, contributing to more resilient and efficient power systems. By providing a solid foundation for future research and development, this study supports the ongoing integration of renewable energy sources and the advancement of modern electrical infrastructure. Full article

In this work, we consider the properties of the two-term Machin-like formula and develop an algorithm for computing digits of $\pi$ by using its rational approximation. In this approximation, both terms are constructed by using a representation of $1/\pi$ in the binary form. This approach provides the squared convergence in computing digits of $\pi$ without any trigonometric functions and surd numbers. The Mathematica codes showing some examples are presented. Full article

We know that the degeneracy of solutions to PDEs, given in terms of theta functions on Riemann surfaces, provides important results about particular solutions, as in the case of the NLS equation. Here, we degenerate the so called finite gap solutions of the Toda lattice equation from the general formulation in terms of abelian functions when the gaps tend to points. This degeneracy allows us to recover the Sato formulas without using inverse scattering theory or geometric or representation theoretic methods. Full article

Devising a financial trading strategy that allows for long-term gains is a very common problem in finance. This paper aims to formulate a mathematically rigorous framework for the problem and compare and contrast the results obtained. The main approach considered is based on Dynamic Bayesian Networks (DBNs). Within the DBN setting, a long-term as well as a short-term trading strategy are considered and applied on twelve equities obtained from developed and developing markets. It is concluded that both the long-term and the medium-term strategies proposed in this paper outperform the benchmark buy-and-hold (B&H) trading strategy. Despite the clear advantages of the former trading strategies, the limitations of this model are discussed along with possible improvements. Full article

We coin the term “network goodness” for a value we define for a network embedded in a given environment as a metric that describes the suitability of that network for meeting a demand. Three formulas are proposed to calculate the metric from three variable values. The first variable considers parts of the environment gravitated by the network. For these parts of the environment, we define a value that measures user costs refusing them the use of the network. Last but not least, the network maintenance costs are considered. The results are obtained after focusing on infrastructure and transport networks, but can be used for other types of networks as well. Full article

Cortical neurons integrate upstream signals and random electrical noise to gate signaling outcomes, leading to statistically random patterns of activity. Yet classically, the neuron is modeled as a binary computational unit, encoding Shannon entropy. Here, the neuronal membrane potential is modeled as a function of inherently probabilistic ion behavior. In this new model, each neuron computes the probability of transitioning from an off-state to an on-state, thereby encoding von Neumann entropy. Component pure states are integrated into a physical quantity of information, and the derivative of this high-dimensional probability distribution yields eigenvalues across the multi-scale quantum system. In accordance with the Hellman–Feynman theorem, the resolution of the system state is paired with a spontaneous shift in charge distribution, so this defined system state instantly becomes the past as a new probability distribution emerges. This mechanistic model produces testable predictions regarding the wavelength of free energy released upon information compression and the temporal relationship of these events to physiological outcomes. Overall, this model demonstrates how cortical neurons might achieve non-deterministic signaling outcomes through a computational process of noisy coincidence detection. Full article