AppliedMath, Volume 5, Issue 1 (March 2025) – 18 articles

The modeling of the one-dimensional wave equation is the fundamental model for characterizing the behavior of vibrating strings in different physical systems. In this work, we investigate numerical solutions for the one-dimensional wave equation employing both explicit and implicit finite difference schemes. To evaluate the correctness of our numerical schemes, we perform extensive error analysis looking at the $L^1$ norm of error and relative error. We conduct thorough convergence tests as we refine the discretization resolutions to ensure that the solutions converge in the correct order of accuracy to the exact analytical solution. Using the von Neumann approach, the stability of the numerical schemes are carefully investigated so that both explicit and implicit schemes maintain the stability criteria over simulations. We test the accuracy of our numerical schemes and present a few examples. We compare the solution with the well-known spectral and finite element method. We also show theoretical proof of the stability and convergence of our numerical scheme. Full article

This study investigates M/M/n:(m/FIFO) systems with a limited queue capacity (incorporating both “waiting and rejection”). This category of systems can be considered to be mixed-service systems. They operate as queuing systems for customers admitted to the system awaiting service, as well as systems that implement rejection or loss for customers who are denied when the system is full (when all servers and the buffer capacity are occupied). The correlation between the system size and a set of performance measures is analysed for the given arrival and service rates. The system size is determined based on a threshold rate of rejected customers. The correlation between the buffer size and the utilisation factor has direct relevance in the design of real systems (e.g., when the dynamics of the arrival rate can be estimated, it provides a solution for phasing the building of physical waiting places for a specific service capacity). In addition, the analysis of customer rejection probability and average waiting time as a function of the effective utilisation factor could yield practical insights for designing and operating real systems. The second part of this study presents a model for optimising the size of a multi-server system with a finite queue capacity. Initially, the number of servers is determined, assuming that the existing situation does not allow for an increase in the buffer capacity. Then, the case in which both server and buffer capacities become decision variables is presented. The operating losses (which are more straightforward to measure than the related costs) are used as an optimisation criterion. Full article

Among the many (mostly randomized) models proposed in the last decades to study how opinions of a set of individuals interconnected by pairwise relations evolve, a novel deterministic model is introduced in this paper that is able to encompass individual choices, strength and sign of relations, and asynchronism. In particular, asynchronism has been considered until now only in randomized settings. It is here studied in which cases the behavior of the resulting dynamical network is predictable, that is, in which cases the number of opinion configurations encountered by the set of individuals before the dynamical network enters a loop is polynomially bounded by the network size. Full article

An efficient linearization method for solving a system of nonlinear equations was developed, showing good stability and convergence properties. It uses an unconventional and simple strategy to improve the performance of classic methods by a full-rank update of the Jacobian approximates. It can be considered both as a discretized Newton’s method or as a quasi-Newton method with a full-rank update of the Jacobian approximates. A solution to the secant equation presented earlier was based on the Wolfe-Popper procedure. The secant equation was splitted into two equations by introducing an auxiliary variable. A simplified algorithm is given in this paper for the full-rank update procedure.It directly solves the secant equation with the pseudoinverse of the Jacobian approximate matrix. Numerical examples are shown for demonstration purposes. The convergence and efficiency of the suggested method are discussed and compared with the convergence and efficiency of classic linearization methods. Full article

The final goal of this paper is to organize the tools needed to study digital Quantum Communications, where classical information is entrusted to quantum states represented by density operators. A density operator is usually defined starting from a set of kets in the Hilbert space and a probability distribution. A fundamental problem in Quantum Communications is the factorization of such operators of the form $\rho =\gamma {\gamma }^{*}$, where $\gamma$ is a matrix called a density factor (DF). The environments considered are finite dimensional Hilbert space (discrete variables) and infinite dimensional Hilbert space (continuous variables). Using discrete variables, the multiplicity and the variety of DFs are investigated using the tools of matrix analysis, arriving in particular to establish the DF with minimal size. With continuous variables, the target is the closed-form factorization, which is achieved with recent results for the important class of Gaussian states. Finally, an application is carried out in Quantum Communications with noisy Gaussian states. Full article

Imaginary coquaternions cℍ can be represented by matrices of negative feedback $N^{-}$, positive feedback $P^{+}$, and reciprocal links $R^{\pm }$. An added environmental element $E^{\pm }$ endows biologic systems with the structure of cℍ module. Although cℍ representation links base patterns with the geometric structure of the pseudo-Euclidean ${\mathbb{R}}_2^4$ space, unknown physiologic aspects of relationships between base elements may add new functional features to the structure of a functional module. Another question is whether achieving and remaining in the equilibrium state provides stability for a biologic system. Considering the property of a biologic system to return deviated conditions to the equilibrium, the system of ordinary differential equations describing the behavior of a mechanical pendulum was modified and used as a basic tool to find the answers. The results obtained show that in evolving systems, the regulatory patterns are organized in a sequence $NPRN$ of base elements, allowing the system to perform a high amount of energy-consuming functions. In order to keep dissipating energy at the same level, the system bifurcates and finalizes its regulatory cycle in $R^{\pm }$ by splitting $P^{+}$ after which the next cycle may begin. Obtained flows are continuous pathways that do not interfere with equilibrium states, thus providing a homeostasis mechanism with nonequilibrium dynamics. Functional transformations reflect changes in the geometry and metric index of the coquaternion. Related coquaternion dynamics show the transformation of a hyperbolic hyperboloid into the closed surface, which is the fusion of the portions of the hyperbolic hyperboloid and two spheres. Full article

Diabetic retinopathy (DR) is a severe microvascular complication of diabetes that affects the eyes, leading to progressive damage to the retina and potential vision loss. Timely intervention and detection are crucial for preventing irreversible damage. With the advancement of technology, deep learning (DL) has emerged as a powerful tool in medical diagnostics, offering a promising solution for the early prediction of DR. This study compares four convolutional neural network architectures, DenseNet201, ResNet50, VGG19, and MobileNetV2, for predicting DR. The evaluation is based on both accuracy and training time data. MobileNetV2 outperforms other models, with a validation accuracy of 78.22%, and ResNet50 has the shortest training time (15.37 s). These findings emphasize the trade-off between model accuracy and computational efficiency, stressing MobileNetV2’s potential applicability for DR prediction due to its balance of high accuracy and a reasonable training time. Performing a 5-fold cross-validation with 100 repetitions, the ensemble of MobileNetV2 and a Graph Convolution Network exhibits a validation accuracy of 82.5%, significantly outperforming MobileNetV2 alone, which shows a 5-fold validation accuracy of 77.4%. This superior performance is further validated by the area under the receiver operating characteristic curve (ROC) metric, demonstrating the enhanced capability of the ensemble method in accurately detecting diabetic retinopathy. This suggests its competence in effectively classifying data and highlights its robustness across multiple validation scenarios. Moreover, the proposed clustering approach can find damaged locations in the retina using the developed Isolate Regions of Interest method, which achieves almost a 90% accuracy. These findings are useful for researchers and healthcare practitioners looking to investigate efficient and effective powerful models for predictive analytics to diagnose diabetic retinopathy. Full article

This article is dedicated to the construction of neural networks for the prediction of the current–voltage characteristic (CVC). CVC is the most important characteristic of the mass transfer process in electro-membrane systems (EMS). CVC is used to evaluate and select the optimal design and effective operating modes of EMS. Each calculation of the CVC at the given values of the input parameters, using developed analytical-numerical models, takes a lot of time, so the CVC is calculated in a limited range of parameter changes. The creation of neural networks allowed for the use of prediction to obtain the CVC for a wider range of input parameter values and much faster, saving computing resources. The regularities of the behavior of CVC for various values of input parameters were revealed. During this work, several different neural network architectures were developed and tested. The best predictive results on test samples are given by the neural network consisting of convolutional and LSTM (Long Short-Term Memory) layers. Full article

The intersection of fractals, non-Euclidean geometry, spatial autocorrelation, and urban structure offers valuable theoretical and practical application insights, which echoes the overarching goal of this paper. Its research question asks about connections between graph theory adjacency matrix eigenfunctions and certain non-Euclidean grid systems; its explorations reflect accompanying synergistic influences on modern urban design. A Minkowski metric with an exponent between one and two bridges Manhattan and Euclidean spaces, supplying an effective tool in these pursuits. This model coalesces with urban fractal dimensions, shedding light on network density and human activity compression. Unlike Euclidean geometry, which assumes unique shortest paths, Manhattan geometry better represents human movements that typically follow multiple equal-length network routes instead of unfettered straight-line paths. Applying these concepts to urban spatial models, like the Burgess concentric ring conceptualization, reinforces the need for fractal analyses in urban studies. Incorporating a fractal perspective into eigenvector methods, particularly those affiliated with spatial autocorrelation, provides a deeper understanding of urban structure and dynamics, enlightening scholars about city evolution and functions. This approach enhances geometric understanding of city layouts and human behavior, offering insights into urban planning, network density, and human activity flows. Blending theoretical and applied concepts renders a clearer picture of the complex patterns shaping urban spaces. Full article

Optimizing learning rates (LRs) in deep learning (DL) has long been challenging. Previous solutions, such as learning rate scheduling (LRS) and adaptive learning rate (ALR) algorithms like RMSProp and Adam, added complexity by introducing new hyperparameters, thereby increasing the cost of model training through expensive cross-validation experiments. These methods mainly focus on local gradient patterns, which may not be effective in scenarios with multiple local optima near the global optimum. A new technique called Learning Rate Tuner with Relative Adaptation (LRT-RA) is introduced to tackle these issues. This approach dynamically adjusts LRs during training by analyzing the global loss curve, eliminating the need for costly initial LR estimation through cross-validation. This method reduces training expenses and carbon footprint and enhances training efficiency. It demonstrates promising results in preventing premature convergence, exhibiting inherent optimization behavior, and elucidating the correlation between dataset distribution and optimal LR selection. The proposed method achieves 84.96% accuracy on the CIFAR-10 dataset while reducing the power usage to 0.07 kWh, $C{O}_2$ emissions to 0.05, and both $S{O}_2$ and $N{O}_x$ emissions to 0.00003 pounds, during the whole training and testing process. Full article

Haberman linking is a widely used method for comparing groups using the two-parameter logistic item response model. However, the traditional Haberman linking approach relies on joint item parameter estimation, which prevents the application of standard M-estimation theory for linking error calculation in the presence of differential item functioning. To address this limitation, a novel pairwise Haberman linking method is introduced. Pairwise Haberman linking aligns with Haberman linking when no items are missing but eliminates the need for joint item parameters, allowing for the use of M-estimation theory in linking error computation. Theoretical derivations and simulation studies show that pairwise Haberman linking delivers reliable statistical inferences for items and persons, particularly in terms of coverage rates. Furthermore, using a bias-corrected linking error is recommended to reduce the influence of sample size on error estimates. Full article

The stock market index future price forecasting is one of the imperative financial time series problems. Accurately estimated future closing prices can play important role in making trading decisions and investment plannings. This work proposes a new multi-output ensemble framework that integrates the hybrid systems generated through importance score based feature weighted learning models through a continuous multi-colony ant colony optimization technique (MACO-LD) for multi-day ahead index future price forecasting. Importance scores are obtained through four different importance score generation strategies (F-test, Relief, Random Forest, and Grey correlation). Multi-output variants of three baseline learning algorithms are brought in to address multi-day ahead forecasting. This study uses three learning algorithms namely multi-output least square support vector regression (MO-LSSVR), multi-output proximal support vector regression (MO-PSVR) and multi-output $\epsilon$-twin support vector regression (MO-$\epsilon$-TSVR) as the baseline methods for the feature weighted hybrid models. For the purpose of forecasting the future price of an index, a comprehensive collection of technical indicators has been taken into consideration as the input features. The proposed study is tested over eight index futures to explore the forecasting performance of individual hybrid predictors obtained after incorporating importance scores over baseline methods. Finally, multi-colony ant colony optimization algorithm is employed to construct the ensemble results from the feature weighted hybrid models along with baseline algorithms. The experimental results for all the eight index futures established that the proposed ensemble of importance score based feature weighted models exhibits superior performance in index future price forecasting compared to the baseline methods and that of importance score based hybrid methods. Full article

We present the derivation of generic equations describing the long gravity waves in incompressible fluid with a decaying effect. We show that in this theory, the only restriction to the surface deviation is connected to the stability condition for the waves. Derivation of these generic equations is based on Euler equations for inviscid incompressible fluid and the definition of dynamic pressure which leads to a correct dispersion equation for gravity waves. These derived generic equations for the velocity of fluid and the surface deviation describe the propagation of long gravity waves in incompressible fluid with finite amplitude. We also find the necessary and sufficient conditions for generic equations with dissipation of energy or a decaying effect. The developed approach can significantly improve the accuracy of theory for long gravity waves in incompressible fluid. We also find the quasi-periodic and solitary wave solutions for generic equations with a decaying effect. Full article

The information paradox suggests that the black hole loses information when it emits radiation. In this way, the spectrum of radiation corresponds to a mixed (non-pure) quantum state even if the internal state generating the black hole is expected to be pure in essence. In this paper we propose an argument solving this paradox by developing an understanding of the process by which spontaneous symmetry breaks when a black hole selects one of the many possible ground states and emits radiation as a consequence of it. Here, the particle operator number is the order parameter. This mechanism explains the connection between the density matrix, corresponding to the pure state describing the black hole state, and the density matrix describing the spectrum of radiation (mixed quantum state). From this perspective, we can recover black hole information from the superposition principle, applied to the different possible order parameters (particle number operators). Full article

According to the latest data from CryptoSlam, as of November 2024, NFT sales have approached USD 7.43 billion, with trading profits exceeding USD 33.303 million. In the buyer–seller market, the potential demand for NFT transactions continues to grow, leading to rapid development in the NFT market and giving rise to various issues, such as price manipulation, counterfeit products, hacking of investment platforms, identity verification errors, data leaks, and wallet security failures, all of which have caused significant financial losses for investors. Currently, the NFT investment market faces challenges such as legal uncertainty, information security, and high price volatility due to speculation. This study conducted expert interviews and adopted a two-stage research methodology to analyze the most common risk factors when selecting NFT investments. It employed the Decision-Making Trial and Evaluation Laboratory (DEMATEL) and the Analytic Network Process (ANP) to explore risk factors such as legal issues, security concerns, speculation, and price volatility, aiming to understand how these factors influence investors in choosing the most suitable NFT investment platform. The survey was conducted between February and June 2023, targeting professionals and scholars with over 10 years of experience in the financial market or financial research, with a total of 13 participants. The empirical results revealed that speculation had the greatest impact compared to legal issues, security concerns, and NFT price volatility. Speculation and price volatility directly influenced other risk factors, potentially increasing the risks faced by NFT investment platforms. In contrast, legal and security issues had less influence on other factors and were more affected by them, indicating a relatively lower likelihood of occurrence. Thus, investors must be cautious of short-term speculation, particularly when dealing with rare NFTs. The best approach is to set an exit price to minimize potential losses if the investment does not proceed as planned. Full article

Given the fundamental importance of the power grid in both supply and demand, frequency stability is critical to the reliable and stable function of energy systems. When energy is stored in the system, it mitigates problems caused by various disturbances that interrupt the energy system’s operation. The energy storage system (ESS) stores excess energy and returns it to the system by reducing power oscillations and improving stability and dependability. Superconducting magnetic energy storage (SMES) is one strategy for storing energy in the power system. As a rotational storage system, its quick dynamic response is a significant advantage. This device can quickly release a substantial amount of energy. A gas power plant in one area, along with a steam and a hydropower plant in another, constitute a multi-resource energy system. This paper’s primary objective is to study and model how SMES affects the dynamic behavior of this energy system. The state-space representation of the power system’s dynamic behavior is given by first-order differential equations. This power system has a complexity of fifteen orders. The outcomes of the simulation using MATLAB software are presented in the time domain, and its correctness is shown by analyzing the power system’s modes. The results show that placing an SMES unit not only eliminates oscillations and frequency deviation but also reduces the induction time in the time responses of power in the connection line and frequency deviation. Different modes are considered for the energy system, and the effect of the power storage unit is shown by presenting the simulation results. Full article