Foundations doi: 10.3390/foundations4040041

Authors: Michael J. W. Hall

Does determinism (or even the incompleteness of quantum mechanics) follow from locality and perfect correlations? In a 1964 paper, John Bell gave the first demonstration that quantum mechanics is incompatible with local hidden variables. Since then, a vigorous debate has rung out over whether he relied on an assumption of determinism or instead, as he later claimed in a 1981 paper, derived determinism from assumptions of locality and perfect correlation. This paper aims to bring clarity to the debate via simple examples and rigorous results. It is first recalled, via quantum and classical counterexamples, that the weakest statistical form of locality consistent with Bell&rsquo;s 1964 paper (parameter independence) is insufficient for the derivation of determinism. Attention is then turned to critically assess Bell&rsquo;s appeal to the Einstein&ndash;Rosen&ndash;Podolsky (EPR) incompleteness argument to support his claim. It is shown that this argument is itself incomplete, via counterexamples that expose two logical gaps. Closing these gaps via a strong &ldquo;counterfactual&rdquo; reality criterion enables a rigorous derivation of both determinism and parameter independence, and in this sense justifies Bell&rsquo;s claim. Conversely, however, it is noted that whereas the EPR argument requires a weaker &ldquo;measurement choice&rdquo; assumption than Bell&rsquo;s demonstration, it nevertheless leads to a similar incompatibility with quantum predictions rather than quantum incompleteness.

]]>Foundations doi: 10.3390/foundations4040040

Authors: Carlos Sosa Henríquez Martynas Lendraitis

This paper presents a comprehensive analytical&ndash;numerical algorithm constructed for proprotor performance evaluation, focusing on accommodating large inflow angles. The algorithm&rsquo;s design, range, and analytical features are clarified, indicating its potential to improve performance analysis, particularly for blades with substantial pitch variations. The Stahlhut model has not been validated against the conventional BEMT small-inflow angle methodology. This paper implements a modified Stahlhut model, coupled with the conventional BEMT. Preliminary validations of the model demonstrate promising results, with deviations reduced to &minus;3% to 4% compared to conventional BEMT methods exhibiting deviations as high as 20% to 88% against experimental data for a highly twisted proprotor. The reconsideration of the computational module carries considerable implications for the design and refinement of proprotors, providing alternative analysis methods that could improve operational effectiveness across a range of flight scenarios. Drawing upon the theoretical framework presented by Stahlhut, the algorithm enables a more complex understanding of proprotor dynamics, facilitating accurate predictions of the loads at each blade section. The introduced algorithm emerges as a valuable asset for evaluating proprotor performance during the early stages of design and certification, offering both low computational cost and medium to high reliability.

]]>Foundations doi: 10.3390/foundations4040039

Authors: Marie-Thérèse Aimar Abdelkader Intissar

In part I of this work to appear in Foudations-MDPI 2024, some existence and uniqueness results for the solutions of some equations were reviewed, such as the Korteweg&ndash;de Vries equation (KdV), the Kuramoto&ndash;Sivashinsky equation (KS), the generalized Korteweg&ndash;de Vries&ndash;Kuramoto&ndash;Sivashinsky equation (gKdV-KS), and the nonhomogeneous boundary value problem for the KdV-KS equation in quarter plane. The main objective of this paper is to review some results of the existence of global attractors for the evolution equations with nonlinearity of the form N(ux), where ux denotes the derivative of u with respect to x, focusing in particular on the Kuramoto&ndash;Sivashinsky equation in one and two dimensions. In order to illustrate the general abstract results, we have chosen to discuss in detail the existence of global attractors for the Kuramoto&ndash;Sivashinsky (KS) equation in 1D and 2D. Once a global attractor is obtained, the question arises whether it has special regularity properties. Then we give an integrated version of the homogeneous steady state Kuramoto&ndash;Sivashinsky equation in Rn. This work ends with a change from rectangular to polar coordinates in the three-dimensional KS equation to give an energy estimate in this case.

]]>Foundations doi: 10.3390/foundations4040038

Authors: Marie-Thérèse Aimar Abdelkader Intissar

This paper reviews the results of existence and uniqueness of the solutions of these equations: the Korteweg&ndash;De Vries equation, the Kuramoto&ndash;Sivashinsky equation, the generalized Korteweg&ndash;De Vries&ndash;Kuramoto&ndash;Sivashinsky equation and the nonhomogeneous boundary value problem for the KdV-KS equation in quarter plane.

]]>Foundations doi: 10.3390/foundations4040037

Authors: Purushottam Das Gujrati

We follow the Boltzmann-Clausius-Maxwell (BCM) proposal to establish the generalized second law (GSL) that is applicable to a system of any size, including a single particle system as our example establishes, and that supercedes the celebrated second law (SL) of increase of entropy of an isolated system. It is merely a consequence of the mechanical equilibrium (stable or unstable) principle (Mec-EQ-P) of analytical mechanics and the first law. We justify an irreversibility priciple that covers all processes, spontaneous or not, and having both positive and negative nonequilibrium temperatures temperatures T defined by (dQ/dS)E. Our novel approach to establish GSL/SL is the inverse of the one used in classical thermodynamics and clarifies the concept of spontaneous processes so that dS&ge;0 for T&gt;0 and dS&lt;0 for T&lt;0. Nonspontaneous processes such as creation of internal constraints are not covered by GSL/SL. Our demonstration establishes that Mec-EQ-P controls spontaneous processes, and that temperature (positive and negative) must be considered an integral part of dissipation.

]]>Foundations doi: 10.3390/foundations4040036

Authors: Dimitris M. Christodoulou Demosthenes Kazanas

We consider spontaneous quark transitions between the &Lambda;0 baryon and its resonant states, and (anti)quark transitions between the neutral kaon K0 and the two heavy &eta;q-mesons (q = c, b). The measured differences in mass deficits are used to calculate the binding energy levels of valence c and b (anti)quarks in these transitions. The method takes into account the isospin energy release in K0 transitions and the work conducted by the strong force in suppressing internal Coulomb repulsions that develop in the charged &Lambda;c+-baryon. We find that the flips s&rarr;c and s&macr;&rarr;c&macr; both release energy back to the strong field and that the overall range of quark energy levels above their u-ground is 100-MeV wider than that of antiquark energy levels above their d&macr;-ground. The wider quark range stems from the flip s&rarr;b, which costs 283 MeV more (or 3&times; more) than the corresponding antiquark flip s&macr;&rarr;b&macr;. At the same time, transitions from the respective ground states to the s and s&macr; states (or the c and c&macr; states) point to a clear origin of the elusive charge-parity (CP) violation. The determined binding energy levels of (anti)quarks allow us to analyze in depth the (anti)quark transitions in &Lambda;-baryons and B-mesons.

]]>Foundations doi: 10.3390/foundations4040035

Authors: Manuel L. Esquível Nadezhda P. Krasii Philippe L. Didier

After reviewing important historical and present day ideas about the concept of time, we develop some instances of mathematical examples where, from the interaction of concepts that model interactions of things in the observable world, time flow emerges in an intuitive and local interpretation. We present several instances of emergence of time flow in mathematical contexts, to wit, by specific parametrisation of deterministic and stochastic curves or of geodesics in Riemann manifolds.

]]>Foundations doi: 10.3390/foundations4040034

Authors: Katharina Lorena Franzke Uwe Gerstmann

Hyperfine splittings play an important role in quantum information and spintronics applications. They allow for the readout of the spin qubits, while at the same time providing the dominant mechanism for the detrimental spin decoherence. Their exact knowledge is thus of prior relevance. In this work, we analytically investigate the relativistic effects on the hyperfine splittings of hydrogen-like atoms, including finite-size effects of the nucleis&rsquo; structure. We start from exact solutions of Dirac&rsquo;s equation using different nuclear models, where the nucleus is approximated by (i) a point charge (Coulomb potential), (ii) a homogeneously charged full sphere, and (iii) a homogeneously charged spherical shell. Equivalent modelling has been done for the distribution of the nuclear magnetic moment. For the 1s ground state and 2s excited state of the one-electron systems H1, H2, H3, and He+3, the calculated finite-size related hyperfine shifts are quite similar for the different structure models and in excellent agreement with those estimated by comparing QED and experiment. This holds also in a simplified approach where relativistic wave functions from a Coulomb potential combined with spherical-shell distributed nuclear magnetic moments promises an improved treatment without the need for an explicit solution of Dirac&rsquo;s equation within the nuclear core. Larger differences between different nuclear structure models are found in the case of the anisotropic 2p3/2 orbitals of hydrogen, rendering these excited states as promising reference systems for exploring the proton structure.

]]>Foundations doi: 10.3390/foundations4040033

Authors: Angelo Morro

This paper addresses the modelling of material behaviour in terms of differential (or rate) equations. To comply with the objectivity principle, recourse is made to invariant fields in the Lagrangian description or to objective time derivatives in the Eulerian description. The thermodynamic consistency is investigated in terms of the Clausius&ndash;Duhem inequality with two unusual features. Firstly, the (non-negative) entropy production is viewed as a constitutive function per se. Secondly, the inequality is viewed as a constraint on the pertinent fields and it is solved by using a representation formula, which allows for the the admissibility of a class of models. For definiteness, models of heat conduction are established, within Lagrangian descriptions, while models of the Navier&ndash;Stokes&ndash;Voigt fluid are investigated within Eulerian descriptions. In connection with thermo-viscous fluids, evolution equations are investigated within the Eulerian description. It is shown that the thermodynamic consistency is compatible with both objective and non-objective evolution equations.

]]>Foundations doi: 10.3390/foundations4040032

Authors: Aghalaya S. Vatsala Govinda Pageni V. Anthony Vijesh

In our article [...]

]]>Foundations doi: 10.3390/foundations4040031

Authors: Hristo Kiskinov Milena Petkova Andrey Zahariev

In the paper by Vatsala et al [...]

]]>Foundations doi: 10.3390/foundations4040030

Authors: Antonella Nannicini Donato Pertici

In this paper, we study centroids, orthocenters, circumcenters, and incenters of geodesic triangles in non-Euclidean geometry, and we discuss the existence of the Euler line in this context. Moreover, we give simple proofs of the existence of a totally geodesic 2-dimensional submanifold containing a given geodesic triangle in the hyperbolic or spherical 3-dimensional geometry.

]]>Foundations doi: 10.3390/foundations4030029

Authors: Jianqiang Zhao

In recent years, a variety of multiple zeta values (MZVs) variants have been defined and studied. One way to produce these variants is to restrict the indices in the definition of MZVs to some fixed parity pattern, which include Hoffman&rsquo;s multiple t-values, Kaneko and Tsumura&rsquo;s multiple T-values, and Xu and this paper&rsquo;s author&rsquo;s multiple S-values. Xu and this paper&rsquo;s author have also considered the so-called multiple mixed values by allowing all possible parity patterns and have studied a few important relations among these values. In this paper, we turn to the finite analogs and the symmetric forms of the multiple mixed values, motivated by a deep conjecture of Kaneko and Zagier, which relates the finite MZVs and symmetric MZVs, and a generalized version of this conjecture by the author to the Euler sum (i.e., level two) setting. We present a few important relations among these values such as the stuffle, reversal, and linear shuffle relations. We also compute explicitly the (conjecturally smallest) generating set in weight one and two cases. In the appendix, we tabulate some dimension computations for various subspaces of the finite multiple mixed values and propose a conjecture.

]]>Foundations doi: 10.3390/foundations4030028

Authors: Jeetendrasingh Maan Benito J. González Emilio R. Negrín

This paper focuses on establishing boundedness properties and Parseval&ndash;Goldstein-type relations for the generalized Mehler&ndash;Fock transform initially introduced by B. L. J. Braaksma and B. M. Meulenbeld (Compositio Math., 18(3):235&ndash;287, 1967). Also, we derive an inversion formula for this transform over Lebesgue spaces.

]]>Foundations doi: 10.3390/foundations4030027

Authors: Barry D. Ganapol

As more and more numerical and analytical solutions to the linear neutron transport equation become available, verification of the numerical results becomes increasingly important. This presentation concerns the development of another benchmark for the linear neutron transport equation in a benchmark series, each employing a different method of solution. In 1D, there are numerous ways of analytically solving the monoenergetic transport equation, such as the Wiener&ndash;Hopf method, based on the analyticity of the solution, the method of singular eigenfunctions, inversion of the Laplace and Fourier transform solutions, and analytical discrete ordinates in the limit, which is arguably one of the most straightforward, to name a few. Another potential method is the PN (Legendre polynomial order N) method, where one expands the solution in terms of full-range orthogonal Legendre polynomials, and with orthogonality and series truncation, the moments form an open set of first-order ODEs. Because of the half-range boundary conditions for incoming particles, however, full-range Legendre expansions are inaccurate near material discontinuities. For this reason, a double PN (DPN) expansion in half-range Legendre polynomials is more appropriate, where one separately expands incoming and exiting flux distributions to preserve the discontinuity at material interfaces. Here, we propose and demonstrate a new method of solution for the DPN equations for an isotropically scattering medium. In comparison to a well-established fully analytical response matrix/discrete ordinate solution (RM/DOM) benchmark using an entirely different method of solution for a non-absorbing 1 mfp thick slab with both isotropic and beam sources, the DPN algorithm achieves nearly 8- and 7-place precision, respectively.

]]>Foundations doi: 10.3390/foundations4030026

Authors: Tomer Shushi

We follow the assumption that relativistic causality is a key element in the structure of quantum mechanics and integrate the speed of light, c, into quantum mechanics through the postulate that the (reduced) Planck constant is a function of c with a leading order of the form &#8463;c&sim;&Lambda;/cp for a constant &Lambda;&gt;0, and p&gt;1. We show how the limit c&rarr;&infin; implies classicality in quantum mechanics and explain why p has to be larger than 1. As the limit c&rarr;&infin; breaks down both relativity theory and quantum mechanics, as followed by the proposed model, it can then be understood through similar conceptual physical laws. We further show how the position-dependent speed of light gives rise to an effective curved space in quantum systems and show that a stronger gravitational field implies higher quantum uncertainties, followed by the varied c. We then discuss possible ways to find experimental evidence of the proposed model using set-ups to test the varying speed of light models and examine analogies of the model based on electrons in semiconductor heterostructures.

]]>Foundations doi: 10.3390/foundations4030025

Authors: Aris Alexopoulos David Neudegg

Methods for controlling electromagnetic fields in materials are presented that mitigate effects such as electrostatic discharge and electromagnetic/radio frequency interference. The first method determines the effective response of composite materials using a d-dimensional effective medium theory. The material consists of inhomogeneous two-layer inclusions with hyperspherical geometry. Non-integer dimensions represent fractal limits. The material medium is composed of a low hypervolume fraction of inclusions that are randomly distributed inside it. The effective response of the dielectric function is obtained using a virial expansion of the Maxwell&ndash;Garnett theory. The other method uses the transformation medium theory and involves the transformation of the material&rsquo;s permittivity and permeability tensors so that the material exhibits a predefined effective response. By selecting appropriate transformations, a homogeneous material medium is transformed into an inhomogeneous version, forcing the electromagnetic fields to propagate along geodesic paths. These geodesics determine the behaviour of the fields inside the material. As a result, the material can be made to exhibit similar physical characteristics as those of a material composed of hyperspherical inclusions. The theoretical analysis presented is further studied and validated via the use of full-wave numerical simulations of Maxwell&rsquo;s equations.

]]>Foundations doi: 10.3390/foundations4030024

Authors: Anthony M. Baudino Harris F. Ciaccio Michael J. Turski Xavier A. Akins Phoebus Sun Cao Elisa Morales Roger D. Sommer Adam R. Johnson Donald J. Wink Kyle A. Grice Kari L. Stone

Zinc is integral to diverse biological functions, acting catalytically, structurally, and supportively in essential enzyme cycles, despite its limited amounts in the body. Targeting zinc enzymes with potent drugs, such as Vorinostat, demonstrates the therapeutic efficacy of zinc-binding ligands, notably in cutaneous T-cell lymphoma treatments. Our study merges experimental and theoretical approaches to analyze the coordination of 8-hydroxylquinoline (8HQ) inhibitors with biomimetic zinc complexes and human histone deacetylase 8 (HDAC8), a monozinc hydrolase enzyme. Assessing 10 8HQ derivatives for structural and electronic characteristics against these models, we observe minimal inhibition efficacy, corroborated through protein&ndash;ligand docking analyses, highlighting the complexities of inhibitor&ndash;zinc enzyme interactions and suggesting intricate noncovalent interactions that are important for ligand binding to enzymes not accounted for in model zinc hydrolase mimics.

]]>Foundations doi: 10.3390/foundations4030023

Authors: Aghalaya S. Vatsala Govinda Pageni

Computation of the solution of the nonlinear Caputo fractional differential equation is essential for using q, which is the order of the derivative, as a parameter. The value of q can be determined to enhance the mathematical model in question using the data. The numerical methods available in the literature provide only the local existence of the solution. However, the interval of existence is known and guaranteed by the natural upper and lower solutions of the nonlinear differential equations. In this work, we develop monotone iterates, together with lower and upper solutions that converge uniformly, monotonically, and quadratically to the unique solution of the Caputo nonlinear fractional differential equation over its entire interval of existence. The nonlinear function is assumed to be the sum of convex and concave functions. The method is referred to as the generalized quasilinearization method. We provide a Caputo fractional logistic equation as an example whose interval of existence is [0,&infin;).

]]>Foundations doi: 10.3390/foundations4030022

Authors: Andreas Schlatter

We explicitly calculate the value of the cosmological constant, &Lambda;, based on the recently developed theory connecting entropic gravity with quantum events induced by transactions, called transactional gravity. We suggest a novel interpretation of the cosmological constant and rigorously show its inverse proportionality to the squared radius of the causal universe &Lambda;~RU&minus;2.

]]>Foundations doi: 10.3390/foundations4030021

Authors: Pedro Ojeda-May Alexander Vergara

The intracellular environment displays complex dynamics influenced by factors such as molecular crowding and the low Reynolds number of the cytoplasm. Enzymes exhibiting active matter properties further heighten this complexity which can lead to memory effects. Molecular simulations often neglect these factors, treating the environment as a &ldquo;thermal bath&rdquo; using the Langevin equation (LE) with white noise. One way to consider these factors is by using colored noise instead within the generalized Langevin equation (GLE) framework, which allows for the incorporation of memory effects that have been observed in experimental data. We investigated the structural and dynamic differences in Shikimate kinase (SK) using LE and GLE simulations. Our results suggest that GLE simulations, which reveal significant changes, could be utilized for assessing conformational motions&rsquo; impact on catalytic reactions.

]]>Foundations doi: 10.3390/foundations4030020

Authors: Edoardo Ballico

We study the multiviews of algebraic space curves X from n pin-hole cameras of a real or complex projective space. We assume the pin-hole centers to be known, i.e., we do not reconstruct them. Our tools are algebro-geometric. We give some general theorems, e.g., we prove that a projective curve (over complex or real numbers) may be reconstructed using four general cameras. Several examples show that no number of badly placed cameras can make a reconstruction possible. The tools are powerful, but we warn the reader (with examples) that over real numbers, just using them correctly, but in a bad way, may give ghosts: real curves which are images of the emptyset. We prove that ghosts do not occur if the cameras are general. Most of this paper is devoted to three important cases of space curves: unions of a prescribed number of lines (using the Grassmannian of all lines in a 3-dimensional projective space), plane curves, and curves of low degree. In these cases, we also see when two cameras may reconstruct the curve, but different curves need different pairs of cameras.

]]>Foundations doi: 10.3390/foundations4020019

Authors: Akihiro Nishiyama Shigenori Tanaka Jack A. Tuszynski

We aim to derive a super-radiance solution of coherent light and sound waves involving water degrees of freedom in the environment of a microtubule. We introduce a Lagrangian density functional of quantum electrodynamics with non-relativistic charged bosons as a model of quantum brain dynamics (QBD) involving water molecular conformational states and photon fields. We also introduce the model of charged boson fields (water degrees of freedom) coupled with phonons. Both optical and acoustic super-radiance solutions are derived in our approach. An acoustic super-radiance mechanism involving information transfer is proposed as an additional candidate to solve the binding problem and to achieve acoustic holography. Our results can be applied to achieve holographic memory storage and information processing in QBD.

]]>Foundations doi: 10.3390/foundations4020018

Authors: Csaba Bús Bence Kutus Áron Ágoston László Janovák Pál Sipos

Alkaline&ndash;surfactant&ndash;polymer (ASP) flooding by means of which alkali additives, surfactant and polymer are inserted as the same slug is one of the most favourable worldwide focuses of Chemical Enhanced Oil Recovery (cEOR) research and field trials, due to the individual synergy of the three chemical components. To develop efficient oil recovery chemicals, it is essential to fully understand the mechanism behind ASP flooding. Nonetheless, there are hardly any studies reporting a systematic characterization of the ASP process. Thus, the present paper focuses on modelling this process in a laboratory by the use of an anionic surfactant&mdash;sodium dodecyl sulphate (SDS) in alkaline&ndash;polymer media&mdash;which is composed of a commercial water-soluble polymer (Flopaam AN125SH&reg;, SNF Floerger, Andr&eacute;zieux-Bouth&eacute;on, France) and alkali compounds (NaOH and Na2CO3). The samples were characterized using rheometry, dynamic light scattering (DLS), infrared spectroscopy (IR) and measurement of inferfacial tension (IFT) between the samples and rapeseed oil. In accordance with the experimental results, surprisingly lower IFT values were recorded between the alkaline&ndash;polymer solutions and rapeseed oil than the samples which contained SDS. Increasing polymer and sodium chloride concentration caused a decrease (from 0.591 mN/m to 0.0486 mN/m) in IFT between the surfactant containing samples and rapeseed oil. The IR measurements confirmed that the surfactant was not detected in the oil phase in the absence of NaOH and Na2CO3. The effects of SDS on the viscosity of the mixtures were also investigated, as viscosity is a considerably important parameter in processes using polymers.

]]>Foundations doi: 10.3390/foundations4020017

Authors: Stefano Innamorati

In the present work, we classify sets of type (4,n) in PG(3,q). We prove that PG(3,q), apart from the planes of PG(3,3), contains only sets of type (4,n) with standard parameters. Thus, somewhat surprisingly, we conclude that there are no sets of type (4,n) in PG(3,q), q &ne; 3, with non-standard parameters.

]]>Foundations doi: 10.3390/foundations4020016

Authors: José Carlos Rosales María Ángeles Moreno-Frías

In this work, we show that if F is a positive integer, then Sat(F)={S&#8739;S is a saturated numerical semigroup with Frobenius number F} is a covariety. As a consequence, we present two algorithms: one that computes Sat(F), and another which computes all the elements of Sat(F) with a fixed genus. If X&sube;S\&Delta;(F) for some S&isin;Sat(F), then we see that there exists the least element of Sat(F) containing X. This element is denoted by Sat(F)[X]. If S&isin;Sat(F), then we define the Sat(F)-rank of S as the minimum of {cardinality(X)&#8739;S=Sat(F)[X]}. In this paper, we present an algorithm to compute all the elements of Sat(F) with a given Sat(F)-rank.

]]>Foundations doi: 10.3390/foundations4020015

Authors: Sean P. Rigby

This work reviews a range of fundamental theoretical considerations in pore structural characterisation. The pore concept is essential for providing a better understanding of physical processes arising within porous media than purely phenomenological approaches. The notion of a pore structure is found to be independently valid and invariant during theory change concerning said physical processes, even for structural models obtained via indirect methods. While imaging methods provide a more direct characterisation of porous solids, there is often a surfeit of information beyond that which can be wielded with current computing power to predict processes sufficiently accurately. Unfortunately, the pore network model extraction methods cannot decide in advance the level of simplification necessary to obtain the optimum minimal idealisation for a given physical process. Pore network models can be obtained with differing geometrical and topological properties, but similar mass transfer rates, for reasons that are often not clear. In contrast, the &lsquo;pore-sifting&rsquo; strategy aims to explicitly identify the key feature of the void space that controls a mass transport process of interest.

]]>Foundations doi: 10.3390/foundations4020014

Authors: Andrew T. Peplow Bilong Liu

This paper introduces an efficient computational procedure for analyzing the propagation of harmonic waves in layered elastic media. This offers several advantages, including the ability to handle arbitrary frequencies, depths, and the number of layers above an elastic half-space, and efforts to follow dispersion curves and flag up possible singularities are investigated. While there are inherent limitations in terms of computational accuracy and capacity, this methodology is straightforward to implement for studying free or forced vibrations and obtaining relevant response data. We present computations of wavenumber dispersion diagrams, phase velocity plots, and response data in both the frequency and time domains. These computational results are provided for two example cases: plane strain and axisymmetry. Our methodology is grounded in a well-conditioned dynamic stiffness approach specifically tailored for deep-layered strata analysis. We introduce an innovative method for efficiently computing wavenumber dispersion curves. By tracking the slope of these curves, users can effectively manage continuation parameters. We illustrate this technique through numerical evidence of a layer resonance in a real-life case study characterized by a fold in the dispersion curves. Furthermore, this framework is particularly advantageous for engineers addressing problems related to ground-borne vibrations. It enables the analysis of phenomena such as zero group velocity (ZGV), where a singularity occurs, both in the frequency and time domains, shedding light on the unique characteristics of such cases. Given the reduced dimension of the problem, this formulation can considerably aid geophysicists and engineers in areas such as MASW or SASW techniques.

]]>Foundations doi: 10.3390/foundations4020013

Authors: David Ellerman

There is a fundamental subsets&ndash;partitions duality that runs through the exact sciences. In more concrete terms, it is the duality between elements of a subset and the distinctions of a partition. In more abstract terms, it is the reverse-the-arrows of category theory that provides a major architectonic of mathematics. The paper first develops the duality between the Boolean logic of subsets and the logic of partitions. Then, probability theory and information theory (as based on logical entropy) are shown to start with the quantitative versions of subsets and partitions. Some basic universal mapping properties in the category of Sets are developed that precede the abstract duality of category theory. But by far the main application is to the clarification and interpretation of quantum mechanics. Since classical mechanics illustrates the Boolean worldview of full distinctness, it is natural that quantum mechanics would be based on the indefiniteness of its characteristic superposition states, which is modeled at the set level by partitions (or equivalence relations). This approach to interpreting quantum mechanics is not a jury-rigged or ad hoc attempt at the interpretation of quantum mechanics but is a natural application of the fundamental duality running throughout the exact sciences.

]]>Foundations doi: 10.3390/foundations4020012

Authors: Antigoni G. Margellou Philippos J. Pomonis

In this work we suggest that the common cause for the development of various power laws is the existence of a suitable exchangeable quantity between the agents of a set. Examples of such exchangeable quantities, leading to eponymous power laws, include money (Pareto&rsquo;s Law), scientific knowledge (Lotka&rsquo;s Law), people (Auerbach&rsquo;s Law), and written or verbal information (Zipf&rsquo;s Law), as well as less common cases like bullets during deadly conflicts, recognition in social networks, heat between the atmosphere and sea-ice floes, and, finally, mass of water vapors between pores in solids. This last case is examined closely in the present article based on extensive experimental data. It is shown that the transferred mass between pores, which eventually grow towards a power law distribution, may be expressed using different parameters, either transferred surface area, or transferred volume, or transferred pore length or transferred pore anisotropy. These distinctions lead to different power laws of variable strength as reflected by the corresponding exponent. The exponents depend quantitatively on the spread of frequency distribution of the examined parameter and tend to zero as the spread of distribution tends to a single order of magnitude. A comparison between the energy and the entropy of different kinds of pore distributions reveals that these two statistical parameters are linearly related, implying that the system poise at a critical state and the exchangeable quantities are the most convenient operations helping to keep this balance.

]]>Foundations doi: 10.3390/foundations4020011

Authors: A. Schlatter R. E. Kastner

The question of where the inertial properties of matter come from has been open for a long time. Isaac Newton considered inertia an intrinsic property of matter. Ernst Mach held a different view whereby the inertia of a body comes from its interaction with the rest of the universe. This idea is known today as Mach&rsquo;s principle. We discuss Mach&rsquo;s principle based on transactional gravity, the recently developed connection of entropic gravity to the physics of quantum events, induced by transactions. It is shown that Mach&rsquo;s principle holds and that there is a fundamental relation between the gravitational constant G and the total mass in the causal universe. This relationship, derived by means of entropic principles, is rigorously proven.

]]>Foundations doi: 10.3390/foundations4020010

Authors: Luca Fabbri

This article considers the Dirac field in polar formulation and shows that when torsion is taken in effective approximation the theory has the thermodynamic properties of a van der Waals gas. It is then shown that in the limit of zero chiral angle the van der Waals gas reduces to a Weyssenhoff fluid, and in spinlessness regime the Weyssenhoff fluid further reduces to a Newton particle. This nesting of approximations allows us to interpret the various spinor quantities. We will see that torsion will provide a form of negative pressure, while the chiral angle will be related to a type of temperature.

]]>Foundations doi: 10.3390/foundations4010009

Authors: Ismael Fernández-Osete David Bermejo Xavier Ayneto-Gubert Xavier Escaler

Nowadays, hydropower plants are being used to compensate for the variable power produced by the new fluctuating renewable energy sources, such as wind and solar power, and to stabilise the grid. Consequently, hydraulic turbines are forced to work more often in off-design conditions, far from their best efficiency point. This new operation strategy increases the probability of erosive cavitation and of hydraulic instabilities and pressure fluctuations that increase the risk of fatigue damage and reduce the life expectancy of the units. To monitor erosive cavitation and fatigue damage, acoustic emissions induced by very-high-frequency elastic waves within the solid have been traditionally used. Therefore, acoustic emissions are becoming an important tool for hydraulic turbine failure detection and troubleshooting. In particular, artificial intelligence is a promising signal analysis research hotspot, and it has a great potential in the condition monitoring of hydraulic turbines using acoustic emissions as a key factor in the digitalisation process. In this paper, a brief introduction of acoustic emissions and a description of their main applications are presented. Then, the research works carried out for cavitation and fracture detection using acoustic emissions are summarised, and the different levels of development are compared and discussed. Finally, the role of artificial intelligence is reviewed, and expected directions for future works are suggested.

]]>Foundations doi: 10.3390/foundations4010008

Authors: Jiong Liu Mohammadamin Dashti Moghaddam Rostislav A. Serota

In this study, we undertake a systematic study of historic market volatility spanning roughly five preceding decades. We focus specifically on the time series of the realized volatility (RV) of the S&amp;P500 index and its distribution function. As expected, the largest values of RV coincide with the largest economic upheavals of the period: Savings and Loan Crisis, Tech Bubble, Financial Crisis and Covid Pandemic. We address the question of whether these values belong to one of the three categories: Black Swans (BS), that is, they lie on scale-free, power-law tails of the distribution; Dragon Kings (DK), defined as statistically significant upward deviations from BS; or Negative Dragons Kings (nDK), defined as statistically significant downward deviations from BS. In analyzing the tails of the distribution with RV&gt;40, we observe the appearance of &ldquo;potential&rdquo; DK, which eventually terminate in an abrupt plunge to nDK. This phenomenon becomes more pronounced with the increase in the number of days over which the average RV is calculated&mdash;here from daily, n=1, to &ldquo;monthly&rdquo;, n=21. We fit the entire distribution with a modified Generalized Beta (mGB) distribution function, which terminates at a finite value of the variable but exhibits a long power-law stretch prior to that, as well as a Generalized Beta Prime (GB2) distribution function, which has a power-law tail. We also fit the tails directly with a straight line on a log-log scale. In order to ascertain BS, DK or nDK behavior, all fits include their confidence intervals and p-values are evaluated for the data points to check whether they can come from the respective distributions.

]]>Foundations doi: 10.3390/foundations4010007

Authors: Chong-En Li Bing-Wen Wu Nae-Wen Kuo Mei-Hua Yuan

Visualizing multivariate data can be challenging, especially for the general public. The difficulties extend beyond determining how to present the data; they also involve comprehension. Early literature has identified various methods, including Chernoff&rsquo;s face, but these methods often have significant drawbacks, making them challenging to interpret. Subsequently, other techniques, such as scatterplots, parallel coordinate plots, and dynamic graphics, have been introduced. However, many of these methods can be intricate to create and interpret, particularly when visualizing high-dimensional data. Additionally, simultaneously representing discrete aspects (including &ldquo;space&rdquo;) and continuous aspects (including &ldquo;time&rdquo;) presents another challenge. This study proposes a novel approach named the &ldquo;&Delta; table&rdquo; (delta table), which transforms space&ndash;time multivariate data consisting of discrete and continuous variables into a tabular format. The &Delta; table is believed to be more user-friendly for the general public, which is its most significant advantage compared to previous methods. Finally, we used a case study of the decoupling of the world&rsquo;s developed, newly industrialized, and developing economies in recent decades as an example of an attempt to apply the &Delta; table.

]]>Foundations doi: 10.3390/foundations4010006

Authors: Wanrong Hong Sili Zhu Jun Li

Natural mathematical objects for representing spatially distributed physical attributes are 3D field functions, which are prevalent in applied sciences and engineering, including areas such as fluid dynamics and computational geometry. The representations of these objects are task-oriented, which are achieved using various techniques that are suitable for specific areas. A recent breakthrough involves using flexible parameterized representations, particularly through neural networks, to model a range of field functions. This technique aims to uncover fields for computational vision tasks, such as representing light-scattering fields. Its effectiveness has led to rapid advancements, enabling the modeling of time dependence in various applications. This survey provides an informative taxonomy of the recent literature in the field of learnable field representation, as well as a comprehensive summary in the application field of visual computing. Open problems in field representation and learning are also discussed, which help shed light on future research.

]]>Foundations doi: 10.3390/foundations4010005

Authors: Manoj K. Singh Ioannis K. Argyros Samundra Regmi

We develop the local convergence of the six order Contraharmonic-mean Newton&rsquo;s method (CHN) to solve Banach space valued equations. Our analysis approach is two fold: The first way uses Taylor&rsquo;s series and derivatives of higher orders. The second one uses only the first derivatives. We examine the theoretical results by solving a boundary value problem also using the examples relating the proposed method with other&rsquo;s methods such as Newton&rsquo;s, Kou&rsquo;s and Jarratt&rsquo;s to show that the proposed method performs better. The conjugate maps for second-degree polynomial are verified. We also calculate the fixed points (extraneous). The article is completed with the study of basins of attraction, which support and further validate the theoretical and numerical results.

]]>Foundations doi: 10.3390/foundations4010004

Authors: Sheikh Zain Majid Muhammad Saeed Umar Ishtiaq Ioannis K. Argyros

Inrecent years, there has been a notable increase in utilising multiple criteria decision-making (MCDM) methods in practical problem solving. The advancement of enhanced decision models with greater capabilities, coupled with technologies like geographic information systems (GIS) and artificial intelligence (AI), has fueled the application of MCDM techniques across various domains. To address the scarcity of irrigation water resources in Bortala, Northwest China, the selection of a dam site has been approached using a hybrid model integrating a multipolar Fuzzy set and a plithogenic Fuzzy hypersoft set along with a GIS. This study considered criteria such as a geological layer, slope, soil type, and land cover. Four potential and reasonably suitable dam locations were identified using a dam construction suitability map developed for Bortala. Ultimately, we showcased the benefits of the innovative method, emphasizing an open, transparent, and science-based approach to selecting optimal dam sites through local studies and group discussions. The results highlight the effectiveness of the hybrid approach involving a fuzzy hypersoft set and plithogenic multipolar fuzzy hypersoft set in addressing the challenges of dam site selection.

]]>Foundations doi: 10.3390/foundations4010003

Authors: Rashed Hasan Ratul Maliha Tasnim Hwang-Cheng Wang Rashadul Hasan Badhon Mohammad Tawhid Kawser

The latest cellular technology, known as 5G-NR, is intended to significantly speed up and improve the effectiveness of wireless systems. A revolution in the telecom industry has been sparked by the widespread use of and increased reliance on cellular communication technology. Moreover, 5G and B5G technologies are expected to utilize an even higher-frequency range to achieve faster data transmission and lower latency communication. Consequently, while transmitting signals across various types of equipment and infrastructure, the general public is exposed to much higher frequencies of electromagnetic radiation. The increasing need for 5G NR base stations (gNodeB) has heightened public anxiety over potential negative health impacts. This study reviews recent research on the effects of electromagnetic waves on humans, particularly focusing on how these effects influence cognitive functions. Most research to date has not found significant differences in cognitive performance due to ubiquitous mobile communications. However, current research has largely been limited to 4G technologies, and the health effects of exposure to 5G user equipment (UE) and base stations in higher-frequency bands remain unexplored. If subsequent research suggests that exposure to high-frequency wireless networks significantly impacts cognitive functions, the deployment and acceptance of these technologies may face challenges and constraints. Therefore, such investigations are crucial for determining whether next-generation technologies pose no risk to individuals.

]]>Foundations doi: 10.3390/foundations4010002

Authors: Petri P. Kärenlampi

Mathematical analysis is conducted on cyclical systems with goodwill in capitalization. Proportional goodwill vanishes with vanishing tangible value. Correspondingly, periodic boundary condition does not enable commercial utilization of the goodwill. Abandoning the periodic boundary condition enables commercial utilization of the goodwill. Even if a physical system is periodic, an agent can abandon the corresponding boundary condition by divesting. Example cases are shown in terms of boreal forestry systems.

]]>Foundations doi: 10.3390/foundations4010001

Authors: Martin Bohner

Foundations published its inaugural issue in 2021, establishing itself as a new international open access, peer-reviewed, multidisciplinary journal of science and techonology, covering mathematics, physics, chemistry, biology, engineering, earth sciences, materials, information sciences, and medical sciences [...]

]]>Foundations doi: 10.3390/foundations3040044

Authors: Eirini S. Papadaki Theodoros Chatzimitakos Vassilis Athanasiadis Dimitrios Kalompatsios Eleni Bozinou Paraskevi Mitlianga Stavros I. Lalas

Pollution of water sources with heavy metals is a pressing environmental issue. To this end, various procedures are being used to remediate water, including sorption. The aim of this study was to investigate the effectiveness of humic acids (HAs) and fulvic acids (FAs) for the removal of metals from water. Specifically, HA and FA were examined for their potential to be used as sorbent materials for 26 heavy metals, alkali metals, and alkaline earth metals. HA and FA were isolated from lignite samples from two mines (Mavropigi mine and South Field mine, Kozani, West Macedonia, Greece). Experiments were carried out using natural mineral water without pH adjustment, so as to gain a better overview of the sorption efficiency in real-life samples. The results showed that FAs were able to sorb most of the examined metals compared to HAs. Several metals such as Ba (34.22&ndash;37.77%), Ca (99.12&ndash;99.58%), and Sr (97.89&ndash;98.12%) were efficiently sorbed when 900 ppm of FAs from both sources were used but were not sorbed by HAs from any source (&le;0.1%). Due to the functional groups on the surface of FA, it is plausible to conclude that it can remove more metals than HA. Meanwhile, lignite from the South Field mine was found to be more efficient for the sorption efficiency in lower concentrations (300&ndash;600 ppm), whereas lignite from the Mavropigi mine was more effective in higher concentrations (900 ppm). For instance, higher removal rates were observed in Mo (62.84%), Pb (56.81%), and U (49.22%) when 300 ppm of HAs of South Field mine were used, whilst the employment of 900 ppm of HAs from Mavropigi mine led to high removal rates of As (49.90%), Se (64.47%), and Tl (85.96%). The above results were also reflected in a principal component analysis, which showed the dispersion of the metal parameters near to or far from the HA and FA parameters depending on their sorption capacity. Overall, both HA and FA could be effectively utilized as sorbent materials for metal removal from water samples. The results of the research indicate a potential application to the remediation of water from metals under dynamic conditions in order to protect public health.

]]>Foundations doi: 10.3390/foundations3040043

Authors: James A. Hoult Yubin Yan

We investigate the spatial discretization of a stochastic semilinear superdiffusion problem driven by fractionally integrated multiplicative space&ndash;time white noise. The white noise is characterized by its properties of being white in both space and time, and the time fractional derivative is considered in the Caputo sense with an order &alpha;&isin; (1, 2). A spatial discretization scheme is introduced by approximating the space&ndash;time white noise with the Euler method in the spatial direction and approximating the second-order space derivative with the central difference scheme. By using the Green functions, we obtain both exact and approximate solutions for the proposed problem. The regularities of both the exact and approximate solutions are studied, and the optimal error estimates that depend on the smoothness of the initial values are established.

]]>Foundations doi: 10.3390/foundations3040042

Authors: Umar Ishtiaq Khaleel Ahmad Farhan Ali Moazzama Faraz Ioannis K. Argyros

Sets, probability, and neutrosophic logic are all topics covered by neutrosophy. Moreover, the classical set, fuzzy set, and intuitionistic fuzzy set are generalized using the neutrosophic set. A neutrosophic set is a mathematical concept used to solve problems with inconsistent, ambiguous, and inaccurate data. In this article, we demonstrate some basic fixed-point theorems for any even number of compatible mappings in complete neutrosophic metric spaces. Our primary findings expand and generalize the findings previously established in the literature.

]]>Foundations doi: 10.3390/foundations3040041

Authors: Charis Anastopoulos Maria-Electra Plakitsi

We analyze time-of-arrival probability distributions for relativistic particles in the context of quantum field theory (QFT). We show that QFT leads to a unique prediction, modulo post-selection that incorporates properties of the apparatus into the initial state. We also show that an experimental distinction of different probability assignments is possible especially in near-field measurements. We also analyze causality in relativistic measurements. We consider a quantum state obtained by a spacetime-localized operation on the vacuum, and we show that detection probabilities are typically characterized by small transient non-causal terms. We explain that these terms originate from Feynman propagation of the initial operation, because the Feynman propagator does not vanish outside the light cone. We discuss possible ways to restore causality, and we argue that this may not be possible in measurement models that involve switching the field&ndash;apparatus coupling on and off.

]]>Foundations doi: 10.3390/foundations3040040

Authors: Muhammad Tariq Sotiris K. Ntouyas Bashir Ahmad

This paper presents an extensive review of some recent results on fractional Ostrowski-type inequalities associated with a variety of convexities and different kinds of fractional integrals. We have taken into account the classical convex functions, quasi-convex functions, (&zeta;,m)-convex functions, s-convex functions, (s,r)-convex functions, strongly convex functions, harmonically convex functions, h-convex functions, Godunova-Levin-convex functions, MT-convex functions, P-convex functions, m-convex functions, (s,m)-convex functions, exponentially s-convex functions, (&beta;,m)-convex functions, exponential-convex functions, &zeta;&macr;,&beta;,&gamma;,&delta;-convex functions, quasi-geometrically convex functions, s&minus;e-convex functions and n-polynomial exponentially s-convex functions. Riemann&ndash;Liouville fractional integral, Katugampola fractional integral, k-Riemann&ndash;Liouville, Riemann&ndash;Liouville fractional integrals with respect to another function, Hadamard fractional integral, fractional integrals with exponential kernel and Atagana-Baleanu fractional integrals are included. Results for Ostrowski-Mercer-type inequalities, Ostrowski-type inequalities for preinvex functions, Ostrowski-type inequalities for Quantum-Calculus and Ostrowski-type inequalities of tensorial type are also presented.

]]>Foundations doi: 10.3390/foundations3040039

Authors: Ioannis K. Argyros Manoj K. Singh Samundra Regmi

We carried out a local comparison between two ninth convergence order schemes for solving nonlinear equations, relying on first-order Fr&eacute;chet derivatives. Earlier investigations require the existence as well as the boundedness of derivatives of a high order to prove the convergence of these schemes. However, these derivatives are not in the schemes. These assumptions restrict the applicability of the schemes, which may converge. Numerical results along with a boundary value problem are given to examine the theoretical results. Both schemes are symmetrical not only in the theoretical results (formation and convergence order), but the numerical and dynamical results are also similar. We calculated the convergence radii of the nonlinear schemes. Moreover, we obtained the extraneous fixed points for the proposed schemes, which are repulsive and are not part of the solution space. Lastly, the theoretical and numerical results are supported by the dynamic results, where we plotted basins of attraction for a selected test function.

]]>Foundations doi: 10.3390/foundations3040038

Authors: Johan Hansson

I. The arena of quantum mechanics and quantum field theory is the abstract, unobserved and unobservable, M-dimensional formal Hilbert space ≠ spacetime. II. The arena of observations—and, more generally, of all events (i.e., everything) in the real physical world—is the classical four-dimensional physical spacetime. III. The “Born rule” is the random process “magically” transforming I into II. Wavefunctions are superposed and entangled only in the abstract space I, never in spacetime II. Attempted formulations of quantum theory directly in real physical spacetime actually constitute examples of “locally real” theories, as defined by Clauser and Horne, and are therefore already empirically refuted by the numerous tests of Bell’s theorem in real, controlled experiments in laboratories here on Earth. Observed quantum entities (i.e., events) are never superposed or entangled as they: (1) exclusively “live” (manifest) in real physical spacetime and (2) are not described by entangled wavefunctions after “measurement” effectuated by III. When separated and treated correctly in this way, a number of fundamental problems and “paradoxes” of quantum theory vs. relativity (i.e., spacetime) simply vanish, such as the black hole information paradox, the infinite zero-point energy of quantum field theory and the quantization of general relativity.

]]>Foundations doi: 10.3390/foundations3030037

Authors: Alexander Robitzsch

Diagnostic classification models (DCMs) are statistical models with discrete latent variables (so-called skills) to analyze multiple binary variables (i.e., items). The one-parameter logistic diagnostic classification model (1PLDCM) is a DCM with one skill and shares desirable measurement properties with the Rasch model. This article shows that the 1PLDCM is indeed a latent class Rasch model. Furthermore, the relationship of the 1PLDCM to extensions of the DCM to mixed, partial, and probabilistic memberships is treated. It is argued that the partial and probabilistic membership models are also equivalent to the Rasch model. The fit of the different models was empirically investigated using six datasets. It turned out for these datasets that the 1PLDCM always had a worse fit than the Rasch model and mixed and partial membership extensions of the DCM.

]]>Foundations doi: 10.3390/foundations3030036

Authors: Alexander B. Kukushkin Andrei A. Kulichenko

The nonlocality (superdiffusion) of turbulence is expressed in the empiric Richardson t3 scaling law for the mean square of the mutual separation of a pair of particles in a fluid or gaseous medium. The development of the theory of nonlocality of various processes in physics and other sciences based on the concept of L&eacute;vy flights resulted in Shlesinger and colleagues&rsquo; about the possibility of describing the nonlocality of turbulence using a linear integro-differential equation with a slowly falling kernel. The approach developed by us made it possible to establish the closeness of the superdiffusion parameter of plasma density fluctuations moving across a strong magnetic field in a tokamak to the Richardson law. In this paper, we show the possibility of a universal description of the characteristics of nonlocality of transfer in a stochastic medium (including turbulence of gases and fluids) using the Biberman&ndash;Holstein approach to examine the transfer of excitation of a medium by photons, generalized in order to take into account the finiteness of the velocity of excitation carriers. This approach enables us to propose a scaling that generalizes Richardson&rsquo;s t3 scaling law to the combined regime of L&eacute;vy flights and L&eacute;vy walks in fluids and gases.

]]>Foundations doi: 10.3390/foundations3030035

Authors: Santhosh George Ioannis Argyros Samundra Regmi

A method without memory as well as a method with memory are developed free of derivatives for solving equations in Banach spaces. The convergence order of these methods is established in the scalar case using Taylor expansions and hypotheses on higher-order derivatives which do not appear in these methods. But this way, their applicability is limited. That is why, in this paper, their local and semi-local convergence analyses (which have not been given previously) are provided using only the divided differences of order one, which actually appears in these methods. Moreover, we provide computable error distances and uniqueness of the solution results, which have not been given before. Since our technique is very general, it can be used to extend the applicability of other methods using linear operators with inverses along the same lines. Numerical experiments are also provided in this article to illustrate the theoretical results.

]]>Foundations doi: 10.3390/foundations3030034

Authors: Sunil Kumar Janak Sharma Ioannis Argyros Samundra Regmi

Derivative-free iterative methods are useful to approximate the numerical solutions when the given function lacks explicit derivative information or when the derivatives are too expensive to compute. Exploring the convergence properties of such methods is crucial in their development. The convergence behavior of such approaches and determining their practical applicability require conducting local as well as semi-local convergence analysis. In this study, we explore the convergence properties of a sixth-order derivative-free method. Previous local convergence studies assumed the existence of derivatives of high order even when the method itself was not utilizing any derivatives. These assumptions imposed limitations on its applicability. In this paper, we extend the local analysis by providing estimates for the error bounds of the method. Consequently, its applicability expands across a broader range of problems. Moreover, the more important and challenging semi-local convergence not investigated in earlier studies is also developed. Additionally, we survey recent advancements in this field. The outcomes presented in this paper can be proved valuable to practitioners and researchers engaged in the development and analysis of derivative-free numerical algorithms. Numerical tests illuminate and validate further the theoretical results.

]]>Foundations doi: 10.3390/foundations3030033

Authors: Samundra Regmi Ioannis Argyros Gagan Deep

Numerous applications from diverse disciplines are formulated as an equation or system of equations in abstract spaces such as Euclidean multidimensional, Hilbert, or Banach, to mention a few. Researchers worldwide are developing methodologies to handle the solutions of such equations. A plethora of these equations are not differentiable. These methodologies can also be applied to solve differentiable equations. A particular method is utilized as a sample via which the methodology is described. The same methodology can be used on other methods utilizing inverses of linear operators. The problem with existing approaches on the local convergence of iterative methods is the usage of Taylor expansion series. This way, the convergence is shown but by assuming the existence of high-order derivatives which do not appear on the iterative methods. Moreover, bounds on the error distances that can be computed are not available in advance. Furthermore, the isolation of a solution of the equation is not discussed either. These concerns reduce the applicability of iterative methods and constitute the motivation for developing this article. The novelty of this article is that it positively addresses all these concerns under weaker convergence conditions. Finally, the more important and harder to study semi-local analysis of convergence is presented using majorizing scalar sequences. Experiments are further performed to demonstrate the theory.

]]>Foundations doi: 10.3390/foundations3030032

Authors: Eugene Oks

There was an error/misprint in the original publication [...]

]]>Foundations doi: 10.3390/foundations3030031

Authors: Valery Astapenko Timur Bergaliyev

A brief review of the classical and quantum description of the interaction of electromagnetic radiation with matter based on the model of a harmonic oscillator is presented. This review includes the generalized Bohr correspondence principle, the excitation of a quantum oscillator by electromagnetic pulses including saturation effect, the harmonic limit of the Bloch equations, and a phenomenological account of the damping of the quantum oscillator. In all cases, at the mathematical level, the relationship between the classical and quantum descriptions of the electromagnetic interaction is established and the conditions for such compliance are identified.

]]>Foundations doi: 10.3390/foundations3030030

Authors: Purushottam Das Gujrati

The review provides a pedagogical but comprehensive introduction to the foundations of a recently proposed statistical mechanics (&mu;NEQT) of a stable nonequilibrium thermodynamic body, which may be either isolated or interacting. It is an extension of the well-established equilibrium statistical mechanics by considering microstates mk in an extended state space in which macrostates (obtained by ensemble averaging A^) are uniquely specified so they share many properties of stable equilibrium macrostates. The extension requires an appropriate extended state space, three distinct infinitessimals d&alpha;=(d,de,di) operating on various quantities q during a process, and the concept of reduction. The mechanical process quantities (no stochasticity) like macrowork are given by A^d&alpha;q, but the stochastic quantities C^&alpha;q like macroheat emerge from the commutator C^&alpha; of d&alpha; and A^. Under the very common assumptions of quasi-additivity and quasi-independence, exchange microquantities deqk such as exchange microwork and microheat become nonfluctuating over mk as will be explained, a fact that does not seem to have been appreciated so far in diverse branches of modern statistical thermodynamics (fluctuation theorems, quantum thermodynamics, stochastic thermodynamics, etc.) that all use exchange quantities. In contrast, dqk and diqk are always fluctuating. There is no analog of the first law for a microstate as the latter is a purely mechanical construct. The second law emerges as a consequence of the stability of the system, and cannot be violated unless stability is abandoned. There is also an important thermodynamic identity&nbsp;diQ&equiv;diW&nbsp;&ge;&nbsp;0 with important physical implications as it generalizes the well-known result of Count Rumford and the Gouy-Stodola theorem of classical thermodynamics. The &mu;NEQT has far-reaching consequences with new results, and presents a new understanding of thermodynamics even of an isolated system at the microstate level, which has been an unsolved problem. We end the review by applying it to three different problems of fundamental interest.

]]>Foundations doi: 10.3390/foundations3030029

Authors: José Bicudo Pedro Nogueira-de-Sá José Chaui-Berlinck

Living beings are composite thermodynamic systems in non-equilibrium conditions. Within this context, there are a number of thermodynamic potential differences (forces) between them and the surroundings, as well as internally. These forces lead to flows, which, ultimately, are essential to life itself, but, at the same time, are associated with entropy generation, i.e., a loss of useful work. The maintenance of homeostatic conditions, the tenet of physiology, demands the regulation of these flows by control of variables. However, due to the very nature of these systems, the regulation of flows and control of variables become entangled in closed loops. Here, we show how to combine entropy generation with respect to a process, and control of parameters (in such a process) in order to create a criterium of optimal ways to regulate changes in flows, the coefficient of flow-entropy (CJσ). We demonstrate the restricted possibility to obtain an increase in flow along with a decrease in entropy generation, and the more general situation of increases in flow along with increases in entropy generation of the process. In this scenario, the CJσ aims to identify the best way to combine the gain in flow and the associated loss of useful work. As an example, we analyze the impact of vaccination effort in the spreading of a contagious disease in a population, showing that the higher the vaccination effort the higher the control over the spreading and the lower the loss of useful work by the society.

]]>Foundations doi: 10.3390/foundations3030028

Authors: Umar Ishtiaq Fahim Din Khaleel Ahmad Doha Kattan Ioannis Argyros

Any two points are close together in a

]]>Foundations doi: 10.3390/foundations3030027

Authors: Juan Núñez Valdés

This article shows the life and work of Gerty Cori, a woman born in Czechoslovakia and who later became a naturalized American, who spent her whole life researching, together with her husband, in the laboratory to find the cause of some diseases, particularly those of a metabolic type, and to be able to find substances that alleviate their effects. The result of this joint work was the obtaining by both, together with the physiologist Bernardo Houssay, of the Nobel Prize in Medicine or Physiology in 1947. The objective of this article is to complete the scarce existing biographies about this woman with new data that highlight the most outstanding events of her life, quite a few of which are still largely ignored. A relatively complete information on the presence of female chemists in the awarded Nobel Prizes is also shown.

]]>Foundations doi: 10.3390/foundations3020026

Authors: Muhammad Tariq Sotiris K. Ntouyas Asif Ali Shaikh

A review of results on Hermite&ndash;Hadamard (H-H) type inequalities in quantum calculus, associated with a variety of classes of convexities, is presented. In the various classes of convexities this includes classical convex functions, quasi-convex functions, p-convex functions,&nbsp;(p,s)-convex functions, modified&nbsp;(p,s)-convex functions,&nbsp;(p,h)-convex functions,&nbsp;tgs-convex functions,&nbsp;&eta;-quasi-convex functions,&nbsp;&#981;-convex functions,&nbsp;(&alpha;,m)-convex functions,&nbsp;&#981;-quasi-convex functions, and coordinated convex functions. Quantum H-H type inequalities via preinvex functions and Green functions are also presented. Finally, H-H type inequalities for&nbsp;(p,q)-calculus, h-calculus, and&nbsp;(q&minus;h)-calculus are also included.

]]>Foundations doi: 10.3390/foundations3020025

Authors: Sotiris K. Ntouyas

The subject of fractional calculus addresses the research of asserted fractional derivatives and integrations over complex domains and their utilization [...]

]]>Foundations doi: 10.3390/foundations3020024

Authors: Qingsong Li Simon Maher

Weber&rsquo;s electrodynamics presents an alternative theory to the widely accepted Maxwell&ndash;Lorentz electromagnetism. It is founded on the concept of direct action between particles, and has recently gained some momentum through theoretical and experimental advancements. However, a major criticism remains: the lack of a comprehensive electromagnetic wave equation for free space. Our motivation in this research article is to address this criticism, in some measure, by deriving an electric wave equation from Weber&rsquo;s electrodynamics based on the axiom of vacuum polarization. Although this assumption has limited experimental evidence and its validity remains a topic of debate among researchers, it has been shown to be useful in the calculation of various quantum mechanical phenomena. Based on this concept, and beginning with Weber&rsquo;s force, we derive an expression which resembles the familiar electric field wave equation derived from Maxwell&rsquo;s equations.

]]>Foundations doi: 10.3390/foundations3020023

Authors: Bernard A. Egwu Yubin Yan

We investigate the application of the Galerkin finite element method to approximate a stochastic semilinear space&ndash;time fractional wave equation. The equation is driven by integrated additive noise, and the time fractional order &alpha;&isin;(1,2). The existence of a unique solution of the problem is proved by using the Banach fixed point theorem, and the spatial and temporal regularities of the solution are established. The noise is approximated with the piecewise constant function in time in order to obtain a stochastic regularized semilinear space&ndash;time wave equation which is then approximated using the Galerkin finite element method. The optimal error estimates are proved based on the various smoothing properties of the Mittag&ndash;Leffler functions. Numerical examples are provided to demonstrate the consistency between the theoretical findings and the obtained numerical results.

]]>Foundations doi: 10.3390/foundations3020022

Authors: Bashir Ahmad Mokhtar Boumaaza Abdelkrim Salim Mouffak Benchohra

In this article, we present some results on the existence and uniqueness of random solutions to a non-linear implicit fractional differential equation involving the generalized Caputo fractional derivative operator and supplemented with non-local and periodic boundary conditions. We make use of the fixed point theorems due to Banach and Krasnoselskii to derive the desired results. Examples illustrating the obtained results are also presented.

]]>Foundations doi: 10.3390/foundations3020021

Authors: Zachary Denton Aghalaya S. Vatsala

One of the key applications of the Caputo fractional derivative is that the fractional order of the derivative can be utilized as a parameter to improve the mathematical model by comparing it to real data. To do so, we must first establish that the solution to the fractional dynamic equations exists and is unique on its interval of existence. The vast majority of existence and uniqueness results available in the literature, including Picard&rsquo;s method, for ordinary and/or fractional dynamic equations will result in only local existence results. In this work, we generalize Picard&rsquo;s method to obtain the existence and uniqueness of the solution of the nonlinear multi-order Caputo derivative system with initial conditions, on the interval where the solution is bounded. The challenge presented to establish our main result is in developing a generalized form of the Mittag&ndash;Leffler function that will cooperate with all the different fractional derivative orders involved in the multi-order nonlinear Caputo fractional differential system. In our work, we have developed the generalized Mittag&ndash;Leffler function that suffices to establish the generalized Picard&rsquo;s method for the nonlinear multi-order system. As a result, we have obtained the existence and uniqueness of the nonlinear multi-order Caputo derivative system with initial conditions in the large. In short, the solution exists and is unique on the interval where the norm of the solution is bounded. The generalized Picard&rsquo;s method we have developed is both a theoretical and a computational method of computing the unique solution on the interval of its existence.

]]>Foundations doi: 10.3390/foundations3020020

Authors: Sotiris K. Ntouyas Bashir Ahmad Jessada Tariboon

In this paper, we study a coupled system of nonlinear proportional fractional differential equations of the Hilfer-type with a new kind of multi-point and integro-multi-strip boundary conditions. Results on the existence and uniqueness of the solutions are achieved by using Banach&rsquo;s contraction principle, the Leray&ndash;Schauder alternative and the well-known fixed-point theorem of Krasnosel&rsquo;ski&#301;. Finally, the main results are illustrated by constructing numerical examples.

]]>Foundations doi: 10.3390/foundations3020019

Authors: Arno Keppens

Building on previous work that considered gravity to emerge from the collective behaviour of discrete, pre-geometric spacetime constituents, this work identifies these constituents with gravitons and rewrites their effective gravity-inducing interaction in terms of local variables for Schwarzschild&ndash;de Sitter scenarios. This formulation enables graviton-level simulations of entire emergent gravitational systems. A first simulation scenario confirms that the effective graviton interaction induces the emergence of spacetime curvature upon the insertion of a graviton condensate into a flat spacetime background. A second simulation scenario demonstrates that free fall can be considered to be fine-tuned towards a geodesic trajectory, for which the graviton flux, as experienced by a test mass, disappears.

]]>Foundations doi: 10.3390/foundations3020018

Authors: Stephen Baghdiguian Emilie Le Goff Laure Paradis Jean Vacelet Nelly Godefroy

The dynamic equilibrium between death and regeneration is well established at the cell level. Conversely, no study has investigated the homeostatic control of shape at the whole organism level through processes involving apoptosis. To address this fundamental biological question, we took advantage of the morphological and functional properties of the carnivorous sponge Lycopodina hypogea. During its feeding cycle, this sponge undergoes spectacular shape changes. Starved animals display many elongated filaments to capture prey. After capture, prey are digested in the absence of any centralized digestive structure. Strikingly, the elongated filaments actively regress and reform to maintain a constant, homeostatically controlled number and size of filaments in resting sponges. This unusual mode of nutrition provides a unique opportunity to better understand the processes involved in cell renewal and regeneration in adult tissues. Throughout these processes, cell proliferation and apoptosis are interconnected key actors. Therefore, L. hypogea is an ideal organism to study how molecular and cellular processes are mechanistically coupled to ensure global shape homeostasis.

]]>Foundations doi: 10.3390/foundations3020017

Authors: Espen Gaarder Haug

The discussion of what matter and mass are has been going on for more than 2500 years. Much has been discovered about mass in various areas, such as relativity theory and modern quantum mechanics. Still, quantum mechanics has not been unified with gravity. This indicates that there is perhaps something essential not understood about mass in relation to gravity. In relation to gravity, several new mass definitions have been suggested in recent years. We will provide here an overview of a series of potential mass definitions and how some of them appear likely preferable for a potential improved understanding of gravity at a quantum level. This also has implications for practical things such as getting gravity predictions with minimal uncertainty.

]]>Foundations doi: 10.3390/foundations3020016

Authors: Jagan Mohan Jonnalagadda

This article establishes a comparison principle for the nabla fractional difference operator &nabla;&rho;(a)&nu;, 1&lt;&nu;&lt;2. For this purpose, we consider a two-point nabla fractional boundary value problem with separated boundary conditions and derive the corresponding Green&rsquo;s function. I prove that this Green&rsquo;s function satisfies a positivity property. Then, I deduce a relatively general comparison result for the considered boundary value problem.

]]>Foundations doi: 10.3390/foundations3020015

Authors: Abhijith Ajayakumar Raju K. George

This paper examines the controllability of a class of heterogeneous networked systems where the nodes are linear time-invariant systems (LTI), and the network topology is triangularizable. The literature contains necessary and sufficient conditions for the controllability of such systems where the control input matrices are identical in each node. Here, we extend this result to a class of heterogeneous systems where the control input matrices are distinct in each node. Additionally, we discuss the controllability of a more general system with triangular network topology and obtain necessary and sufficient conditions for controllability. Theoretical results are supplemented with numerical examples.

]]>Foundations doi: 10.3390/foundations3020014

Authors: Samundra Regmi Ioannis Argyros Gagan Deep Laxmi Rathour

The present paper includes the local and semilocal convergence analysis of a fourth-order method based on the quadrature formula in Banach spaces. The weaker hypotheses used are based only on the first Fréchet derivative. The new approach provides the residual errors, number of iterations, convergence radii, expected order of convergence, and estimates of the uniqueness of the solution. Such estimates are not provided in the approaches using Taylor expansions involving higher-order derivatives, which may not exist or may be very expensive or impossible to compute. Numerical examples, including a nonlinear integral equation and a partial differential equation, are provided to validate the theoretical results.

]]>Foundations doi: 10.3390/foundations3010013

Authors: Samundra Regmi Ioannis K. Argyros Jinny Ann John Jayakumar Jayaraman

Iterative methods which have high convergence order are crucial in computational mathematics since the iterates produce sequences converging to the root of a non-linear equation. A plethora of applications in chemistry and physics require the solution of non-linear equations in abstract spaces iteratively. The derivation of the order of the iterative methods requires expansions using Taylor series formula and higher-order derivatives not present in the method. Thus, these results cannot prove the convergence of the iterative method in these cases when such higher-order derivatives are non-existent. However, these methods may still converge. Our motivation originates from the need to handle these problems. No error estimates are given that are controlled by constants. The process introduced in this paper discusses both the local and the semi-local convergence analysis of two step fifth and multi-step 5+3r order iterative methods obtained using only information from the operators on these methods. Finally, the novelty of our process relates to the fact that the convergence conditions depend only on the functions and operators which are present in the methods. Thus, the applicability is extended to these methods. Numerical applications complement the theory.

]]>Foundations doi: 10.3390/foundations3010012

Authors: Ioannis Argyros Samundra Regmi Jinny John Jayakumar Jayaraman

High-convergence order iterative methods play a major role in scientific, computational and engineering mathematics, as they produce sequences that converge and thereby provide solutions to nonlinear equations. The convergence order is calculated using Taylor Series extensions, which require the existence and computation of high-order derivatives that do not occur in the methodology. These results cannot, therefore, ensure that the method converges in cases where there are no such high-order derivatives. However, the method could converge. In this paper, we are developing a process in which both the local and semi-local convergence analyses of two related methods of the sixth order are obtained exclusively from information provided by the operators in the method. Numeric applications supplement the theory.

]]>Foundations doi: 10.3390/foundations3010011

Authors: Petri P. Kärenlampi

Multiannual growth systems are modeled in generic terms and investigated using partial derivatives and Lagrange multipliers. Grown stock density and temperature sum are used as independent variables. Estate capitalization increases continuously with grown stock and temperature sum, whereas capital return rate and gross profit rate reach a maximum with respect to grown stock. As two restrictions are applied simultaneously, the results mostly but not always follow intuition. The derivative of capital return rate with respect to gross profit rate is positive, and negative with respect to capitalization. The derivative of capitalization with respect to capital return rate shows some positive values, as well as that with respect to gross profit rate. The derivative of the gross profit rate is positive with respect to both capitalization and capital return rate. The results indicate a variety of alternative strategies, which may or may not be multiobjective.

]]>Foundations doi: 10.3390/foundations3010010

Authors: Iván Lechuga Karo Michaelian

Theories on life&rsquo;s origin generally acknowledge the advantage of a semi-permeable vesicle (protocell) for enhancing the chemical reaction&ndash;diffusion processes involved in abiogenesis. However, more and more evidence indicates that the origin of life is concerned with the photo-chemical dissipative structuring of the fundamental molecules under soft UV-C light (245&ndash;275 nm). In this paper, we analyze the Mie UV scattering properties of such a vesicle created with long-chain fatty acids. We find that the vesicle could have provided early life with a shield from the faint but destructive hard UV-C ionizing light (180&ndash;210 nm) that probably bathed Earth&rsquo;s surface from before the origin of life and at least until 1200 million years after, until the formation of a protective ozone layer as a result of the evolution of oxygenic photosynthesis.

]]>Foundations doi: 10.3390/foundations3010009

Authors: Ioannis Argyros Gagan Deep Samundra Regmi

In this study, we present a convergence analysis of a Newton-like midpoint method for solving nonlinear equations in a Banach space setting. The semilocal convergence is analyzed in two different ways. The first one is shown by replacing the existing conditions with weaker and tighter continuity conditions, thereby enhancing its applicability. The second one uses more general ω-continuity conditions and the majorizing principle. This approach includes only the first order Fréchet derivative and is applicable for problems that were otherwise hard to solve by using approaches seen in the literature. Moreover, the local convergence is established along with the existence and uniqueness region of the solution. The method is useful for solving Engineering and Applied Science problems. The paper ends with numerical examples that show the applicability of our convergence theorems in cases not covered in earlier studies.

]]>Foundations doi: 10.3390/foundations3010008

Authors: Valentina Verdoliva Michele Saviano Stefania De Luca

Zeolites, both natural and synthetic, are certainly some of the most versatile minerals for their applications. Since the 1940s, they have been used in the chemical industry as catalysts, adsorbents and ion exchanger extensively, and the development of their practical usage is expected to continue upon years. Their versatility is the result of the combination of peculiar and indispensable properties, each of which can be found in other material as a single property, but seldom all of them are found in combination. However, despite the success of their employment, the mechanisms of many important catalytic processes involving zeolites remained elusive. In particular, the comprehension of the structure&ndash;property relationships for emerging applications are highly required. In this perspective article we focus on the role of zeolites as solid acid-base catalysts. We go deeply into the structural properties of the LTA kind (Zeolite-Na A 4 &Aring;ngstrom) that was successfully employed as basic catalyst for several nucleophilic substitution reactions.

]]>Foundations doi: 10.3390/foundations3010007

Authors: Eugene Oks

There exists the following paradigm: for interaction potentials U(r) that are negative and go to zero as r goes to infinity, bound states may exist only for the negative total energy E. For E &gt; 0 and for E = 0, bound states are considered to be impossible, both in classical and quantum mechanics. In the present paper we break this paradigm. Namely, we demonstrate the existence of bound states of E = 0 in neutron&ndash;neutron systems and in neutron&ndash;muon systems, specifically when the magnetic moments of the two particles in the pair are parallel to each other. As particular examples, we calculate the root-mean-square size of the bound states of these systems for the values of the lowest admissible values of the angular momentum, and show that it exceeds the neutron radius by an order of magnitude. We also estimate the average kinetic energy and demonstrate that it is nonrelativistic. The corresponding bound states of E = 0 may be called &ldquo;neutronium&rdquo; (for the neutron&ndash;neutron systems) and &ldquo;neutron&ndash;muonic atoms&rdquo; (for the neutron&ndash;muon systems). We also point out that this physical system possesses higher-than-geometric (i.e., algebraic) symmetry, leading to the approximate conservation of the square of the angular momentum, despite the geometric symmetry being axial. We use this fact for facilitating analytical and numerical calculations.

]]>Foundations doi: 10.3390/foundations3010006

Authors: Foundations Editorial Office Foundations Editorial Office

High-quality academic publishing is built on rigorous peer review [...]

]]>Foundations doi: 10.3390/foundations3010005

Authors: Henok Desalegn Desta Eze R. Nwaeze Tadesse Abdi Jebessa B. Mijena

In this paper, by using Jensen&ndash;Mercer&rsquo;s inequality we obtain Hermite&ndash;Hadamard&ndash;Mercer&rsquo;s type inequalities for a convex function employing left-sided (k,&nbsp;&psi;)-proportional fractional integral operators involving continuous strictly increasing function. Our findings are a generalization of some results that existed in the literature.

]]>Foundations doi: 10.3390/foundations3010004

Authors: Ahmed M. A. El-Sayed Yasmin M. Y. Omar Hind H. G. Hashem Shorouk M. Al-Issa

This article is devoted to the solvability and the asymptotic stability of a coupled system of a functional integral equation on the real half-axis. Our consideration is located in the space of bounded continuous functions on R+(BC(R+)). The main tool applied in this work is the technique associated with measures of noncompactness in BC(R+) by a given modulus of continuity. Next, we formulate and prove a sufficient condition for the solvability of that coupled system. We, additionally, provide an example and some particular cases to demonstrate the effectiveness and value of our results.

]]>Foundations doi: 10.3390/foundations3010003

Authors: Gagan Deep Ioannis K. Argyros

In the present study, two new compositions of convergence order six are presented for solving nonlinear equations. The first method is obtained from the third-order one given by Homeier using linear interpolation, and the second one is obtained from the third-order method given by Traub using divided differences. The first method requires three evaluations of the function and one evaluation of the first derivative, thereby enhancing the efficiency index. In the second method, the computation of a derivative is reduced by approximating it using divided differences. Various numerical experiments are performed which demonstrate the accuracy and efficacy of the proposed methods.

]]>Foundations doi: 10.3390/foundations3010002

Authors: László Nemes Christian G. Parigger

This work communicates cavity ring-down spectroscopy (CRDS) of methylidyne (CH) in a chemiluminescent plasma that is produced in a microwave cavity. Of interest are the rotational lines of the 0-0 vibrational transition for the A&ndash;X band and the 1-0 vibrational transition for the B&ndash;X band. The reported investigations originate from research on the CH radical in 1996, which constituted the first case of applying CRDS to the CH radical. The report also includes a recent analysis that shows excellent agreement of the measured and computed data, and it communicates CH line strength data. The CH radical is an important diatomic molecule in hydrocarbon combustion diagnosis and the analysis of stellar plasma emissions, to name just two examples of analytical plasma chemistry.

]]>Foundations doi: 10.3390/foundations3010001

Authors: Christian G. Parigger

This work communicates line-strength data and associated scripts for the computation and spectroscopic fitting of selected transitions of diatomic molecules. The scripts for data analysis are designed for inclusion in various software packages or program languages. Selected results demonstrate the applicability of the program for data analysis in laser-induced optical breakdown spectroscopy primarily at the University of Tennessee Space Institute, Center for Laser Applications. Representative spectra are calculated and referenced to measured data records. Comparisons of experiment data with predictions from other tabulated diatomic molecular databases confirm the accuracy of the communicated line-strength data.

]]>Foundations doi: 10.3390/foundations2040074

Authors: Aghalaya S. Vatsala Govinda Pageni V. Anthony Vijesh

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&lt;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&rsquo;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&rarr;1.

]]>Foundations doi: 10.3390/foundations2040073

Authors: Boris M. Smirnov

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&rsquo;s surface at a rate of about 10&minus;4&#8728;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&rsquo;s thermal state are associated with changes in the Earth&rsquo;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 &#8728;C according to measurements, we construct the energetic balance for each of Earth&rsquo;s states. The fast transition between the Earth&rsquo;s cold and warm states results from the change of the Earth&rsquo;s albedo due to the different volcano activity in these states. There is the feedback between the aggregate state of water covering the Earth&rsquo;s surface and volcanic eruptions, which become intense when ice covers approximately 40% of the Earth&rsquo;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.

]]>Foundations doi: 10.3390/foundations2040072

Authors: Nita H. Shah Nisha Sheoran

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&ndash;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.

]]>Foundations doi: 10.3390/foundations2040071

Authors: Kusal Rathnayake Alexander Lebedev Dimitri Volchenkov

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.

]]>Foundations doi: 10.3390/foundations2040070

Authors: J. Gerard Wolff

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 &lsquo;dirty data&rsquo;; 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.

]]>Foundations doi: 10.3390/foundations2040069

Authors: Ioannis K. Argyros Samundra Regmi Christopher I. Argyros Debasis Sharma

We compare the convergence balls and the dynamical behaviors of two efficient weighted-Newton-like equation solvers by Sharma and Arora, and Grau-S&aacute;nchez 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&ndash;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-S&aacute;nchez 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-S&aacute;nchez et al. has larger convergence balls than the solver of Sharma and Arora. Hence, the solver presented by Grau-S&aacute;nchez 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.

]]>Foundations doi: 10.3390/foundations2040068

Authors: Ioannis K. Argyros Christopher I. Argyros Jinny Ann John Jayakumar Jayaraman

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.

]]>Foundations doi: 10.3390/foundations2040067

Authors: Sebastián Michel-Mata Mónica Gómez-Salazar Víctor Castaño Iván Santamaría-Holek

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.

]]>Foundations doi: 10.3390/foundations2040066

Authors: Tejmani Kumar Prashant K. Rai Abhishek K. Rai Nilesh K. Rai Awadhesh K. Rai Christian G. Parigger Geeta Watal Suman Yadav

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.

]]>Foundations doi: 10.3390/foundations2040065

Authors: Christof Baumgärtel Simon Maher

This article reviews the electrodynamic force law of Wilhelm Weber and its importance in electromagnetic theory. An introduction is given to Weber&rsquo;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&rsquo;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&rsquo;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.

]]>Foundations doi: 10.3390/foundations2040064

Authors: Christian G. Parigger

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 &mu;s, 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 &mu;s. 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 &mu;s. 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 &mu;s 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 &mu;s, but are masked by spectra of N2 early in the plasma decay.

]]>Foundations doi: 10.3390/foundations2040063

Authors: Ayub Samadi Sotiris K. Ntouyas Bashir Ahmad Jessada Tariboon

This paper is concerned with the existence of solutions for a new boundary value problem of nonlinear coupled (k,&psi;)&ndash;Hilfer fractional differential equations subject to coupled (k,&psi;)&ndash;Riemann&ndash;Liouville fractional integral boundary conditions. We prove two existence results by applying the Leray&ndash;Schauder alternative, and Krasnosel&rsquo;ski&#301;&rsquo;s fixed-point theorem under different criteria, while the third result, concerning the uniqueness of solutions for the given problem, relies on the Banach&rsquo;s contraction mapping principle. Examples are included for illustrating the abstract results.

]]>Foundations doi: 10.3390/foundations2040062

Authors: Eugene Oks

The proton radius puzzle is one of the most fundamental challenges of modern physics. Before the year 2010, the proton charge radius rp 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 rp from the experiment on muonic hydrogen atoms and they claimed rp 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 rp as from the muonic hydrogen experiments, while others reconfirmed the larger value of rp. 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.

]]>Foundations doi: 10.3390/foundations2040061

Authors: Jeet Amrit Pattnaik Joshua T. Majekodunmi Mrutunjaya Bhuyan Suresh Kumar Patra

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&ndash;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 Br&uuml;ckner energy density functional, and its surface, as well as volume components, are evaluated within Danielwicz&rsquo;s liquid drop prescription. In addition, a comparison between Br&uuml;ckner 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 206Hg. The formation of the kink is traceable to the early filling of the 1i11/2 orbitals rather than 2g9/2, 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.

]]>Foundations doi: 10.3390/foundations2040060

Authors: Paul W. Eloe Jeffrey T. Neugebauer

We construct a Green&rsquo;s function for the three-term fractional differential equation &minus;D0+&alpha;u+aD0+&mu;u+f(t)u=h(t), 0&lt;t&lt;b, where &alpha;&isin;(2,3], &mu;&isin;(1,2], and f is continuous, satisfying the boundary conditions u(0)=u&prime;(0)=0, D0+&beta;u(b)=0, where &beta;&isin;[0,2]. To accomplish this, we first construct a Green&rsquo;s function for the two-term problem &minus;D0+&alpha;u+aD0+&mu;u=h(t), 0&lt;t&lt;b, satisfying the same boundary conditions. A lemma from spectral theory is integral to our construction. Some limiting properties of the Green&rsquo;s function for the two-term problem are also studied. Finally, existence results are given for a nonlinear problem.

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