AppliedMath, Volume 4, Issue 2 (June 2024) – 14 articles

We develop a framework for creating distortion functions that are used to construct new bivariate copulas. It is achieved by transforming non-negative random variables with Lomax-related distributions. In this paper, we apply the distortions to the base copulas of independence, Clayton, Frank, and Gumbel copulas. The properties of the tail dependence coefficient, tail order, and concordance ordering are explored for the new families of distorted copulas. We conducted an empirical study using the daily net returns of Amazon and Google stocks from January 2014 to December 2023. We compared the popular Clayton, Gumbel, Frank, and Gaussian copula models to their corresponding distorted copula models induced by the unit-Lomax and unit-inverse Pareto distortions. The new families of distortion copulas are equipped with additional parameters inherent in the distortion function, providing more flexibility, and are demonstrated to perform better than the base copulas. After analyzing the data, we have found that the joint extremes of Amazon and Google stocks are more likely for high daily net returns than for low daily net returns. Full article

The main purpose of the present paper is to prove the existence of periodic solutions of the three-body problem in the 3D Kepler formulation. We have solved the same problem in the case when the three particles are considered in an external inertial system. We start with the three-body equations of motion, which are a subset of the equations of motion (previously derived by us) for any number of bodies. In the Minkowski space, there are 12 equations of motion. It is proved that three of them are consequences of the other nine, so their number becomes nine, as much as the unknown trajectories are. The Kepler formulation assumes that one particle (the nucleus) is placed at the coordinate origin. The motion of the other two particles is described by a neutral system with respect to the unknown velocities. The state-dependent delays arise as a consequence of the finite vacuum speed of light. We obtain the equations of motion in spherical coordinates and split them into two groups. In the first group all arguments of the unknown functions are delays. We take their solutions as initial functions. Then, the equations of motion for the remaining two particles must be solved to the right of the initial point. To prove the existence–uniqueness of a periodic solution, we choose a space consisting of periodic infinitely smooth functions satisfying some supplementary conditions. Then, we use a suitable operator which acts on these spaces and whose fixed points are periodic solutions. We apply the fixed point theorem for the operators acting on the spaces of periodic functions. In this manner, we show the stability of the He atom in the frame of classical electrodynamics. In a previous paper of ours, we proved the existence of spin functions for plane motion. Thus, we confirm the Bohr and Sommerfeld’s hypothesis for the He atom. Full article

The infinite series solution to the boundary-value problems of Laplace’s equation with discontinuous Dirichlet boundary conditions was found by using the basic method of separation of variables. The merit of this paper is that the closed-form solution, or the singular similarity solution in the semi-infinite strip domain and the first-quadrant domain, can be generated from the basic infinite series solution in the rectangular domain. Moreover, based on the superposition principle, the infinite series solution in the rectangular domain can be related to the singular similarity solution in the semi-infinite strip domain. It is proven that the analytical source-type singular behavior in the infinite series solution near certain singular points in the rectangular domain can be revealed from the singular similarity solution in the semi-infinite strip domain. By extending the boundary of the rectangular domain, the infinite series solution to Laplace’s equation in the first-quadrant domain can be derived to obtain the analytical singular similarity solution in a direct and much easier way than by using the methods of Fourier transform, images, and conformal mapping. Full article

The G-modified Helmholtz equation is a partial differential equation that enables us to express gravity intensity g as a series of spherical harmonics having radial distance r in irrational powers. The Laplace equation in three-dimensional space (in Cartesian coordinates, is the sum of the second-order partial derivatives of the unknown quantity equal to zero) is used to express the Earth’s gravity potential (disturbing and normal potential) in order to represent other useful quantities—which are also known as functionals of the disturbing potential—such as gravity disturbance, gravity anomaly, and geoid undulation as a series of spherical harmonics. We demonstrate that by using the G-modified Helmholtz equation, not only gravity intensity but also disturbing potential and its functionals can be expressed as a series of spherical harmonics. Having gravity intensity represented as a series of spherical harmonics allows us to create new Global Gravity Models. Furthermore, a more detailed examination of the Earth’s isogravitational surfaces is conducted. Finally, we tabulate our results, which makes it clear that new Global Gravity Models for gravity intensity g will be very useful for many geophysical and geodetic applications. Full article

In our study, we investigate the phenomenon of information loss, as measured by the Kullback–Leibler divergence, in a many-fermion system, such as the Lipkin model. Information loss is introduced as the number N of particles increases, particularly when the system is in a mixed state. We find that there is a significant loss of information under these conditions. However, we observe that this loss nearly disappears when the system is in a pure state. Our analysis employs tools from information theory to quantify and understand these effects. Full article

This paper presents a novel deep-learning network designed to detect intervals of jump discontinuities in single-variable piecewise smooth functions from their noisy samples. Enhancing the accuracy of jump discontinuity estimations can be used to find a more precise overall approximation of the function, as traditional approximation methods often produce significant errors near discontinuities. Detecting intervals of discontinuities is relatively straightforward when working with exact function data, as finite differences in the data can serve as indicators of smoothness. However, these smoothness indicators become unreliable when dealing with highly noisy data. In this paper, we propose a deep-learning network to pinpoint the location of a jump discontinuity even in the presence of substantial noise. Full article

The threat posed by the COVID-19 pandemic has been accompanied by an epidemic of misinformation, causing confusion and mistrust among the public. Misinformation about COVID-19 whether intentional or unintentional takes many forms, including conspiracy theories, false treatments, and inaccurate information about the origins and spread of the virus. Though the pandemic has brought to light the significant impact of misinformation on public health, mathematical modeling emerged as a valuable tool for understanding the spread of COVID-19 and the impact of public health interventions. However, there has been limited research on the mathematical modeling of the spread of misinformation related to COVID-19. In this paper, we present a mathematical model of the spread of misinformation related to COVID-19. The model highlights the challenges posed by misinformation, in that rather than focusing only on the reproduction number that drives new infections, there is an additional threshold parameter that drives the spread of misinformation. The equilibria of the model are analyzed for both local and global stability, and numerical simulations are presented. We also discuss the model’s potential to develop effective strategies for combating misinformation related to COVID-19. Full article

We study properties of the minimal Terracini loci, i.e., families of certain zero-dimensional schemes, in a projective plane. Among the new results here are: a maximality theorem and the existence of arbitrarily large gaps or non-gaps for the integers x for which the minimal Terracini locus in degree d is non-empty. We study similar theorems for the critical schemes of the minimal Terracini sets. This part is framed in a more general framework. Full article

The transition zone (TZ) of hydrocarbon reservoirs is an integral part of the hydrocarbon pool which contains a substantial fraction of the deposit, particularly in carbonate petroleum systems. Consequently, knowledge of its thickness and petrophysical properties, viz. its pore size distribution and wettability characteristic, is critical to optimizing hydrocarbon production in this zone. Using classical formation evaluation techniques, the thickness of the transition zone has been estimated, using well logging methods including resistivity and Nuclear Magnetic Resonance, among others. While hydrocarbon fluids’ accumulation in petroleum reservoirs occurs due to the migration and displacement of originally water-filled potential structural and stratigraphic traps, the development of their TZ integrates petrophysical processes that combine spontaneous capillary imbibition and wettability phenomena. In the literature, wettability phenomena have been shown to also be governed by electrostatic phenomena. Therefore, given that reservoir rocks are aggregates of minerals with ionizable surface groups that facilitate the development of an electric double layer, a definite theoretical relationship between the TZ and electrostatic theory must be feasible. Accordingly, a theoretical approach to estimating the TZ thickness, using the electrostatic theory and based on the electric double layer theory, is attractive, but this is lacking in the literature. Herein, we fill the knowledge gap by using the interfacial electrostatic theory based on the fundamental tenets of the solution to the Poisson–Boltzmann mean field theory. Accordingly, we have used an existing model of capillary rise based on free energy concepts to derive a capillary rise equation that can be used to theoretically predict observations based on the TZ thickness of different reservoir rocks, using well-established formation evaluation methods. The novelty of our work stems from the ability of the model to theoretically and accurately predict the TZ thickness of the different lithostratigraphic units of hydrocarbon reservoirs, because of the experimental accessibility of its model parameters. Full article

The sum of independent circular uniformly distributed random variables is also circular uniformly distributed. In this study, it is shown that a family of circular distributions based on nonnegative trigonometric sums (NNTS) is also closed under summation. Given the flexibility of NNTS circular distributions to model multimodality and skewness, these are good candidates for use as alternative models to test for circular uniformity to detect different deviations from the null hypothesis of circular uniformity. The circular uniform distribution is a member of the NNTS family, but in the NNTS parameter space, it corresponds to a point on the boundary of the parameter space, implying that the regularity conditions are not satisfied when the parameters are estimated by using the maximum likelihood method. Two NNTS tests for circular uniformity were developed by considering the standardised maximum likelihood estimator and the generalised likelihood ratio. Given the nonregularity condition, the critical values of the proposed NNTS circular uniformity tests were obtained via simulation and interpolated for any sample size by the fitting of regression models. The validity of the proposed NNTS circular uniformity tests was evaluated by generating NNTS models close to the circular uniformity null hypothesis. Full article

The new approach to quantum mechanics (QM) is that the mathematics of QM is the linearization of the mathematics of partitions (or equivalence relations) on a set. This paper develops those ideas using vector spaces over the field ${\mathbb{Z}}_2=\left\{0.1\right\}$ as a pedagogical or toy model of (finite-dimensional, non-relativistic) QM. The $0,1$-vectors are interpreted as sets, so the model is “quantum mechanics over sets” or QM/Sets. The key notions of partitions on a set are the logical-level notions to model distinctions versus indistinctions, definiteness versus indefiniteness, or distinguishability versus indistinguishability. Those pairs of concepts are the key to understanding the non-classical ‘weirdness’ of QM. The key non-classical notion in QM is the notion of superposition, i.e., the notion of a state that is indefinite between two or more definite- or eigen-states. As Richard Feynman emphasized, all the weirdness of QM is illustrated in the double-slit experiment, so the QM/Sets version of that experiment is used to make the key points. Full article

The frequency deviation from the nominal working frequency in power systems is a consequence of the imbalance between total electrical loads and the aggregate power supplied by production units. The sensitivity of energy system frequency to both minor and major load variations underscore the need for effective frequency load control mechanisms. In this paper, frequency load control in single-area power system with multi-source energy is analysed and simulated. Also, the effect of the photovoltaic system on the frequency deviation changes in the energy system is shown. In the single area energy system, the dynamics of thermal turbine with reheat, thermal turbine without reheat and hydro turbine are considered. The simulation results using Simulink/Matlab and model analysis using eigenvalue analysis show the dynamic behaviour of the power system in response to changes in the load. Full article

In this paper, an analytical exact form of the ramp function is presented. This seminal function constitutes a fundamental concept of the digital signal processing theory and is also involved in many other areas of applied sciences and engineering. In particular, the ramp function is performed in a simple manner as the pointwise limit of a sequence of real and continuous functions with pointwise convergence. This limit is zero for strictly negative values of the real variable $x$, whereas it coincides with the independent variable $x$ for strictly positive values of the variable $x$. Here, one may elucidate beforehand that the pointwise limit of a sequence of continuous functions can constitute a discontinuous function, on the condition that the convergence is not uniform. The novelty of this work, when compared to other research studies concerning analytical expressions of the ramp function, is that the proposed formula is not exhibited in terms of miscellaneous special functions, e.g., gamma function, biexponential function, or any other special functions, such as error function, hyperbolic function, orthogonal polynomials, etc. Hence, this formula may be much more practical, flexible, and useful in the computational procedures, which are inserted into digital signal processing techniques and other engineering practices. Full article

Accurate forecasting of the coronavirus disease 2019 (COVID-19) spread is indispensable for effective public health planning and the allocation of healthcare resources at all levels of governance, both nationally and globally. Conventional prediction models for the COVID-19 pandemic often fall short in precision, due to their reliance on homogeneous time-dependent transmission rates and the oversight of geographical features when isolating study regions. To address these limitations and advance the predictive capabilities of COVID-19 spread models, it is imperative to refine model parameters in accordance with evolving insights into the disease trajectory, transmission rates, and the myriad economic and social factors influencing infection. This research introduces a novel hybrid model that combines classic epidemic equations with a recurrent neural network (RNN) to predict the spread of the COVID-19 pandemic. The proposed model integrates time-dependent features, namely the numbers of individuals classified as susceptible, infectious, recovered, and deceased (SIRD), and incorporates human mobility from neighboring regions as a crucial spatial feature. The study formulates a discrete-time function within the infection component of the SIRD model, ensuring real-time applicability while mitigating overfitting and enhancing overall efficiency compared to various existing models. Validation of the proposed model was conducted using a publicly available COVID-19 dataset sourced from Italy. Experimental results demonstrate the model’s exceptional performance, surpassing existing spatiotemporal models in three-day ahead forecasting. This research not only contributes to the field of epidemic modeling but also provides a robust tool for policymakers and healthcare professionals to make informed decisions in managing and mitigating the impact of the COVID-19 pandemic. Full article