Foundations, Volume 2, Issue 4 (December 2022) – 19 articles

It is known that, from a modeling point of view, fractional dynamic equations are more suitable compared to integer derivative models. In fact, a fractional dynamic equation is referred to as an equation with memory. To demonstrate that the fractional dynamic model is better than the corresponding integer model, we need to compute the solutions of the fractional differential equations and compare them with an integer model relative to the data available. In this work, we will illustrate that the linear $nq$-order sequential Caputo fractional differential equations, which are sequential of order q where $q<1$ with fractional initial conditions and/or boundary conditions, can be solved. The reason for choosing sequential fractional dynamic equations is that linear non-sequential Caputo fractional dynamic equations with constant coefficients cannot be solved in general. We used the Laplace transform method to solve sequential Caputo fractional initial value problems. We used fractional boundary conditions to compute Green’s function for sequential boundary value problems. In addition, the solution of the sequential dynamic equations yields the solution of the corresponding integer-order differential equations as a special case as $q\to 1$. Full article

The evolution of the atmospheric temperature in the past, resulted from the EPICA project (European Project for Ice Coring in Antarctica) for the analysis of air bubbles in ice deposits near three weather stations in Antarctica, includes several glacial cycles. According to these studies, the glacial cycle consists of a slow cooling of the Earth’s surface at a rate of about ${10}^{-4}$${}^{\circ }$C per year for almost the entire time of a single cycle (about 100 thousand years) and of a fast process of heating the planet, similar to a thermal explosion. The observed cooling of the planet follows from the imbalance of energy fluxes absorbed by the Earth and going into its surrounding space, and this imbalance is four orders of magnitude less than the accuracy of determination of the fluxes themselves. The inconsistency of the popular Milankovich theory is shown, according to which glacial cycles in the evolution of the Earth’s thermal state are associated with changes in the Earth’s orbit relative to the Sun. In considering the glacial cycle as the transition between the warm (contemporary) and cold thermal states of the Earth with a difference in their temperatures of 12 ${}^{\circ }$C according to measurements, we construct the energetic balance for each of Earth’s states. The fast transition between the Earth’s cold and warm states results from the change of the Earth’s albedo due to the different volcano activity in these states. There is the feedback between the aggregate state of water covering the Earth’s surface and volcanic eruptions, which become intense when ice covers approximately $40\%$ of the Earth’s surface. Dust measurements in ice deposits within the framework of the EPICA project confirms roughly a heightened volcano eruption during the cold phase of the glacial cycle. Numerical parameters of processes related to the glacial cycle are analyzed. Full article

It is well known that HIV (human immunodeficiency virus) weakens the immune system of individuals, resulting in risk of other infections, such as pneumonia. The most frequent viral pneumonia seen in individuals infected with HIV is cytomegalovirus (CMV). In this paper, pneumonia–HIV co-infection is modeled through the formulation of a mathematical compartmental model consisting of nine compartments. Some of the basic properties of the model are established, such as the positivity, boundedness of the system, equilibrium points, and computation of the basic reproduction number. After obtaining the solution, the homotopy perturbation method (HPM) is applied, as it is known for its convergence properties. It is observed that the HPM gives an accurate analytical solution that indicates various important factors that are responsible for the spread of cytomegalovirus pneumonia in HIV-infected populations, and this is justified through a plot made by using MATLAB 2020a. Full article

A psychology experiment examining decision-making on a continuum of subjectively equivalent alternatives (directions) revealed that subjects follow a common pattern, giving preference to just a few directions over all others. When restricted experimental settings made the common pattern unfeasible, subjects demonstrated no common choice preferences. In the latter case, the observed distribution of choices made by a group of subjects was close to normal. We conclude that the abundance of subjectively equivalent alternatives may reduce the individual variability of choices, and vice versa. Choice overload paradoxically results in behavior patterning and eventually facilitates decision predictability, while restricting the range of available options fosters individual variability of choice, reflected in almost random behavior across the group. Full article

This paper highlights 20 significant problems in AI research, with potential solutions via the SP Theory of Intelligence (SPTI) and its realisation in the SP Computer Model. With other evidence referenced in the paper, this is strong evidence in support of the SPTI as a promising foundation for the development of human-level broad AI, aka artificial general intelligence. The 20 problems include: the tendency of deep neural networks to make major errors in recognition; the need for a coherent account of generalisation, over- and under-generalisation, and minimising the corrupting effect of ‘dirty data’; how to achieve one-trial learning; how to achieve transfer learning; the need for transparency in the representation and processing of knowledge; and how to eliminate the problem of catastrophic forgetting. In addition to its promise as a foundation for the development of AGI, the SPTI has potential as a foundation for the study of human learning, perception, and cognition. And it has potential as a foundation for mathematics, logic, and computing. Full article

We compare the convergence balls and the dynamical behaviors of two efficient weighted-Newton-like equation solvers by Sharma and Arora, and Grau-Sánchez et al. First of all, the results of ball convergence for these algorithms are established by employing generalized Lipschitz constants and assumptions on the first derivative only. Consequently, outcomes for the radii of convergence, measurable error distances and the existence–uniqueness areas for the solution are discussed. Then, the complex dynamical behaviors of these solvers are compared by applying the attraction basin tool. It is observed that the solver suggested by Grau-Sánchez et al. has bigger basins than the method described by Sharma and Arora. Lastly, our ball analysis findings are verified on application problems and the convergence balls are compared. It is found that the method given by Grau-Sánchez et al. has larger convergence balls than the solver of Sharma and Arora. Hence, the solver presented by Grau-Sánchez et al. is more suitable for practical application. The convergence analysis uses the first derivative in contrast to the aforementioned studies, utilizing the seventh derivative not on these methods. The developed process can be used on other methods in order to increase their applicability. Full article

We propose the semi-local convergence of two derivative-free, competing methods of order six to address non-linear equations. The sufficient convergence criteria are the same, making a direct comparison between them possible. The existing convergence technique uses the standard Taylor series approach, which requires derivatives up to order seven. The novelty and originality of our work lies in the fact that in contrast to previous research works, our convergence theorems only demand the first derivative. In addition, formulas for determining the region of uniqueness for solution, convergence radii, and error estimations are suggested. Such results cannot be found in works relying on the seventh derivatives. As a consequence, we are able to broaden the utility of these productive methods. The confirmation of our convergence findings through application problems brings this research to a close. Full article

An innovative and integrative modeling strategy for assessing the sustainability and resilience of social-ecological systems (SES) is presented by introducing a social-ecological entropy production (SEEP) method. In analogy to the thermodynamic entropy production of irreversible processes, we discuss a theoretical model that relates energy and information flow with the cultural and epistemological peculiarities of different communities that exploit the same natural resource. One of the innovative aspects of our approach comes from the fact that sustainability is assessed by a single parameter (SEEP) incorporating the simulation outcomes of all the populations participating in the dynamics, and not only on the fate of the resource. This is significant as far as the non-linearities introduced by the coupling of the different dynamics considered may lead to high sensitivity to small perturbations. Specifically, by assuming two possible types of technical and environmental knowledge-transfer methods [direct (D) and phase-in (P)] within each one of the two communities that exploit and restore a resource, we generate four mathematical models to explore the long-term sustainability scenario due to the intervention, by a new epistemological community, of an initially sustainable resource-community SES. By exploring the space of four key parameters characterizing the degree of technical and environmental knowledge, as well as the rates of social inclusion and knowledge transfer, our simulations show that, from 400 scenarios studied in each case, the P-P model predicts 100% sustainable cases in the use of the resource after the intervention by the second community. The mixed scenarios P-D and D-P predict about 29%, and the D-D scenario only predicts 23% of sustainable cases. Catastrophic outcomes are predicted at about 71% in P-D and D-P scenarios, and about 77% of extinction of the system by exhaustion of the resource and community populations in the D-D scenario. In this form, our theoretical strategy and the knowledge-transfer scenarios studied may help policymakers to find a priori science-based criteria to solve possible controversies arising from social-ecological interventions. Full article

This interdisciplinary work communicates the identification and quantification of elements responsible for the bioactive potency of leaves from pointed gourd, trichosanthes dioica, using laser-induced breakdown spectroscopy (LIBS). Calibration-free LIBS determines the presence of various trace and major elements, their concentrations, and ratios in which they are present in the leaves. The presence of specific elemental ratios of magnesium/sodium and magnesium/potassium could be promising for managing diabetes mellitus. Variable doses of aqueous extract from trichosanthes dioica leaves are administered for determination of the most effective one. Based on encouraging results, the extract could be harvested to serve as anti-diabetic medication for diabetes and associated symptoms. Full article

This article reviews the electrodynamic force law of Wilhelm Weber and its importance in electromagnetic theory. An introduction is given to Weber’s force and it is shown how it has been utilised in the literature to explain electromagnetism as well as phenomena in other disciplines of physics, where the force law has connections to the nuclear force, gravity, cosmology, inertia and quantum mechanics. Further, criticism of Weber’s force is reviewed and common misconceptions addressed and rectified. It is found that, while the theory is not without criticism and has much room for improvement, within the limitations of its validity, it is equally as successful as Maxwell’s theory in predicting certain phenomena. Moreover, it is discussed how Weber offers a valid alternative explanation of electromagnetic phenomena which can enrich and complement the field perspective of electromagnetism through a particle based approach. Full article

This work investigates spatial and temporal distributions of hydroxyl, OH, in laser-plasma in laboratory air at standard ambient temperature and pressure. Of interest are determination of temperature and density of OH and establishment of a correlation of molecular OH emission spectra with shadow graphs for time delays of 50 to 100 μs, analogous to previous work on shadow graph and emission spectroscopy correlation for cyanide, CN, in gas mixtures and for time delays of the order of 1 μs. Wavelength- and sensitivity-corrected spatiotemporal data analysis focuses on temperature inferences using molecular OH emission spectroscopy. Near-IR radiation from a Q-switched laser device initiates optical breakdown in laboratory air. The laser device provides 6 ns, up to 850 milli Joule, pulses at a wavelength of 1064 nm, and focal irradiance in the range of 1 to 10 terawatt per centimeter-squared. Frequency doubled beams are utilized for capturing shadow graphs for visualization of the breakdown kernel at time delays in the range of 0.1 to 100 μs. OH emission spectra of the laser plasma, spatially resolved along the slit dimension, are recorded in the wavelength range of 298 nm to 321 nm, and with gate widths adjusted to 10 μs for the intensified charge-coupled device that is mounted at the exit plane of a 0.64 m Czerny-Turner configuration spectrometer. Diatomic OH signals occur due to recombination of the plasma and are clearly distinguishable for time delays larger than 50 μs, but are masked by spectra of N₂ early in the plasma decay. Full article

This paper is concerned with the existence of solutions for a new boundary value problem of nonlinear coupled $(k,\psi )$–Hilfer fractional differential equations subject to coupled $(k,\psi )$–Riemann–Liouville fractional integral boundary conditions. We prove two existence results by applying the Leray–Schauder alternative, and Krasnosel’skiĭ’s fixed-point theorem under different criteria, while the third result, concerning the uniqueness of solutions for the given problem, relies on the Banach’s contraction mapping principle. Examples are included for illustrating the abstract results. Full article

The proton radius puzzle is one of the most fundamental challenges of modern physics. Before the year 2010, the proton charge radius r_p was determined by the spectroscopic method, relying on the electron energy levels in hydrogen atoms, and by the elastic scattering of electrons on protons. In 2010, and then in 2013, two research teams determined r_p from the experiment on muonic hydrogen atoms and they claimed r_p to be by about 4% smaller than it was found from the experiments with electronic hydrogen atoms. Since then, several research groups performed corresponding experiments with electronic hydrogen atoms and obtained contradictory results: some of them claimed that they found the same value of r_p as from the muonic hydrogen experiments, while others reconfirmed the larger value of r_p. The conclusion of the latest papers (including reviews) is that the puzzle is not resolved yet. In the present paper, we bring to the attention of the research community, dealing with the proton radius puzzle, the contributing factor never taken into account in any previous calculations. This factor has to do with the hydrogen atoms of the second flavor, whose existence is confirmed in four different types of atomic experiments. We present a relatively simple model illustrating the role of this factor. We showed that disregarding the effect of even a relatively small admixture of the second flavor of muonic hydrogen atoms to the experimental gas of muonic hydrogen atoms could produce the erroneous result that the proton charge radius is by about 4% smaller than its actual value, so that the larger out of the two disputed values of the proton charge radius could be, in fact, correct. Full article

The present study is focused on revealing a characteristic kink of the neutron shell closure N = 126 across the Hg-isotopic chain within the relativistic mean-field (RMF) approach with the IOPB-I, DD-ME2, DD-PC1 and NL3 parameter sets. The RMF densities are converted to their spherical equivalence via the Wood–Saxon approximation and used as input within the parametrization procedure of the coherent density fluctuation model (CDFM). The nuclear matter symmetry energy is calculated using the Brückner energy density functional, and its surface, as well as volume components, are evaluated within Danielwicz’s liquid drop prescription. In addition, a comparison between Brückner and relativistic energy density functionals using the NL3 parameter set is shown as a representative case. The binding energy, charge distribution radius and symmetry energy are used as indicators of the isotopic shift in both ground and isomeric states. We have found the presence of a kink at the shell/sub-shell closure at N = 126 for neutron-rich ${}^{206}$Hg. The formation of the kink is traceable to the early filling of the 1$i_{11/2}$ orbitals rather than 2$g_{9/2}$, due to the large spin-orbit splitting. As such, the link between the occupational probability and the magicity of nuclei over the Hg-isotopic chain is established. Full article

We construct a Green’s function for the three-term fractional differential equation $-D_{0+}^{\alpha }u+a{D}_{0^{+}}^{\mu }u+f\left(t\right)u=h\left(t\right)$, $0<t<b$, where $\alpha \in (2,3]$, $\mu \in (1,2]$, and f is continuous, satisfying the boundary conditions $u\left(0\right)=u^{\prime }\left(0\right)=0$, $D_{0^{+}}^{\beta }u\left(b\right)=0$, where $\beta \in [0,2]$. To accomplish this, we first construct a Green’s function for the two-term problem $-D_{0+}^{\alpha }u+a{D}_{0^{+}}^{\mu }u=h\left(t\right)$, $0<t<b$, satisfying the same boundary conditions. A lemma from spectral theory is integral to our construction. Some limiting properties of the Green’s function for the two-term problem are also studied. Finally, existence results are given for a nonlinear problem. Full article

In this paper, we treat some fractional differential equations on the sequence Lebesgue spaces ${\ell }^p\left({\mathbb{N}}_0\right)$ with $p\ge 1$. The Caputo fractional calculus extends the usual derivation. The operator, associated to the Cauchy problem, is defined by a convolution with a sequence of compact support and belongs to the Banach algebra ${\ell }^1\left(\mathbb{Z}\right)$. We treat in detail some of these compact support sequences. We use techniques from Banach algebras and a Functional Analysis to explicity check the solution of the problem. Full article

The double nature of material particles, i.e., their wave and corpuscular characteristics, is usually considered incomprehensible, as it cannot be represented visually. It is proposed to the student, in introductory courses, as a fact justified by quantum interference experiments for which, however, no further analysis is possible. On this note, we propose a description of the wave function in terms of a simple electrical analogy, which reproduces at least some of its essential properties. Our aim is to provide a cognitive representation of an analogical type: starting from a classical context (electrical circuits) and introducing in an appropriate way the notions of “wave” and “particle”, we show how typically quantum properties such as delocalization and entanglement emerge in a natural, understandable, and intuitive way. Full article

We consider a predictor–corrector numerical method for solving Caputo–Hadamard fractional differential equation over the uniform mesh $\log t_j=\log a+\left(\log \frac{t_N}{a}\right)\left(\frac{j}{N}\right),j=0,1,2,\dots ,N$ with $a\ge 1$, where $\log a=\log t_0<\log t_1<\dots <\log t_N=\log T$ is a partition of $[\log a,\log T]$. The error estimates under the different smoothness properties of the solution y and the nonlinear function f are studied. Numerical examples are given to verify that the numerical results are consistent with the theoretical results. Full article

The semi-local convergence is presented for a one parameter seventh order method to obtain solutions of Banach space valued nonlinear models. Existing works utilized hypotheses up to the eighth derivative to prove the local convergence. But these high order derivatives are not on the method and they may not exist. Hence, the earlier results can only apply to solve equations containing operators that are at least eight times differentiable although this method may converge. That is why, we only apply the first derivative in our convergence result. Therefore, the results on calculable error estimates, convergence radius and uniqueness region for the solution are derived in contrast to the earlier proposals dealing with the less challenging local convergence case. Hence, we enlarge the applicability of these methods. The methodology used does not depend on the method and it is very general. Therefore, it can be used to extend other methods in an analogous way. Finally, some numerical tests are performed at the end of the text, where the convergence conditions are fulfilled. Full article