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Foundations, Volume 2, Issue 1 (March 2022) – 22 articles

Cover Story (view full-size image): A plethora of problems from many disciplines can be reduced to solving nonlinear Banach space-valued equations. Most solution methods are of an iterative nature since closed-form solutions are possible only in special cases. We study the semi-local convergence of a three-step Newton-type method for solving nonlinear equations under the classical Lipschitz conditions for first-order derivatives. Earlier studies have used convergence analysis requiring the existence of the sixth derivative not appearing on the method, hence restricting the applicability of the method to operators which are at least six times differentiable. View this paper.
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Review
Non-Equilibrium Thermodynamic Foundations of the Origin of Life
Foundations 2022, 2(1), 308-337; https://doi.org/10.3390/foundations2010022 - 21 Mar 2022
Cited by 2 | Viewed by 1342
Abstract
There is little doubt that life’s origin followed from the known physical and chemical laws of Nature. The most general scientific framework incorporating the laws of Nature and applicable to most known processes to good approximation, is that of thermodynamics and its extensions [...] Read more.
There is little doubt that life’s origin followed from the known physical and chemical laws of Nature. The most general scientific framework incorporating the laws of Nature and applicable to most known processes to good approximation, is that of thermodynamics and its extensions to treat out-of-equilibrium phenomena. The event of the origin of life should therefore also be amenable to such an analysis. In this review paper, I describe the non-equilibrium thermodynamic foundations of the origin of life for the non-expert from the perspective of the “Thermodynamic Dissipation Theory for the Origin of Life” which is founded on Classical Irreversible Thermodynamic theory developed by Lars Onsager, Ilya Prigogine, and coworkers. A Glossary of Thermodynamic Terms can be found at the end of the article to aid the reader. Full article
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Article
Generalized Fractional Integrals Involving Product of a Generalized Mittag–Leffler Function and Two H-Functions
Foundations 2022, 2(1), 298-307; https://doi.org/10.3390/foundations2010021 - 11 Mar 2022
Viewed by 1059
Abstract
The objective of this research is to obtain some fractional integral formulas concerning products of the generalized Mittag–Leffler function and two H-functions. The resulting integral formulas are described in terms of the H-function of several variables. Moreover, we give some illustrative [...] Read more.
The objective of this research is to obtain some fractional integral formulas concerning products of the generalized Mittag–Leffler function and two H-functions. The resulting integral formulas are described in terms of the H-function of several variables. Moreover, we give some illustrative examples for the efficiency of the general approach of our results. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
Article
A Note on a Coupled System of Hilfer Fractional Differential Inclusions
Foundations 2022, 2(1), 290-297; https://doi.org/10.3390/foundations2010020 - 03 Mar 2022
Cited by 1 | Viewed by 920
Abstract
A coupled system of Hilfer fractional differential inclusions with nonlocal integral boundary conditions is considered. An existence result is established when the set-valued maps have non-convex values. We treat the case when the set-valued maps are Lipschitz in the state variables and we [...] Read more.
A coupled system of Hilfer fractional differential inclusions with nonlocal integral boundary conditions is considered. An existence result is established when the set-valued maps have non-convex values. We treat the case when the set-valued maps are Lipschitz in the state variables and we avoid the applications of fixed point theorems as usual. An illustration of the results is given by a suitable example. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
Article
Fundamental Spacetime Representations of Quantum Antenna Systems
Foundations 2022, 2(1), 251-289; https://doi.org/10.3390/foundations2010019 - 02 Mar 2022
Viewed by 1118
Abstract
We utilize relativistic quantum mechanics to develop general quantum field-theoretic foundations suitable for understanding, analyzing, and designing generic quantum antennas for potential use in secure quantum communication systems and other applications. Quantum antennas are approached here as abstract source systems capable of producing [...] Read more.
We utilize relativistic quantum mechanics to develop general quantum field-theoretic foundations suitable for understanding, analyzing, and designing generic quantum antennas for potential use in secure quantum communication systems and other applications. Quantum antennas are approached here as abstract source systems capable of producing what we dub “quantum radiation.” We work from within a generic relativistic framework, whereby the quantum antenna system is modeled in terms of a fundamental quantum spacetime field. After developing a framework explaining how quantum radiation can be understood using the methods of perturbative relativistic quantum field theory (QFT), we investigate in depth the problem of quantum radiation by a controlled abstract source functions. We illustrate the theory in the case of the neutral Klein-Gordon linear quantum antenna, outlining general methods for the construction of the Green’s function of a source—receiver quantum antenna system, the latter being useful for the computation of various candidate angular quantum radiation directivity and gain patterns analogous to the corresponding concepts in classical antenna theory. We anticipate that the proposed formalism may be extended to deal with a large spectrum of other possible controlled emission types for quantum communications applications, including, for example, the production of scalar, fermionic, and bosonic particles, where each could be massless or massive. Therefore, our goal is to extend the idea of antenna beyond electromagnetic waves, where now our proposed QFT-based concept of a quantum antenna system could be used to explore scenarios of controlled radiation of any type of relativistic particles, i.e., effectively transcending the well-known case of photonic systems through the deployment of novel non-standard quantum information transmission carriers such as massive photons, spin-1/2 particles, gravitons, antiparticles, higher spin particles, and so on. Full article
(This article belongs to the Special Issue Advances in Fundamental Physics)
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Article
On the Local Convergence of a (p + 1)-Step Method of Order 2p + 1 for Solving Equations
Foundations 2022, 2(1), 242-250; https://doi.org/10.3390/foundations2010018 - 20 Feb 2022
Cited by 1 | Viewed by 1049
Abstract
The local convergence of a generalized (p+1)-step iterative method of order 2p+1 is established in order to estimate the locally unique solutions of nonlinear equations in the Banach spaces. In earlier studies, convergence analysis for [...] Read more.
The local convergence of a generalized (p+1)-step iterative method of order 2p+1 is established in order to estimate the locally unique solutions of nonlinear equations in the Banach spaces. In earlier studies, convergence analysis for the given iterative method was carried out while assuming the existence of certain higher-order derivatives. In contrast to this approach, the convergence analysis is carried out in the present study by considering the hypothesis only on the first-order Fréchet derivatives. This study further provides an estimate of convergence radius and bounds of the error for the considered method. Such estimates were not provided in earlier studies. In view of this, the applicability of the given method clearly seems to be extended over the wider class of functions or problems. Moreover, the numerical applications are presented to verify the theoretical deductions. Full article
(This article belongs to the Special Issue Iterative Methods with Applications in Mathematical Sciences)
Article
On the Semi-Local Convergence of a Jarratt-Type Family Schemes for Solving Equations
Foundations 2022, 2(1), 234-241; https://doi.org/10.3390/foundations2010017 - 17 Feb 2022
Viewed by 1014
Abstract
We study semi-local convergence of two-step Jarratt-type method for solving nonlinear equations under the classical Lipschitz conditions for first-order derivatives. To develop a convergence analysis we use the approach of restricted convergence regions in combination to majorizing scalar sequences and our technique of [...] Read more.
We study semi-local convergence of two-step Jarratt-type method for solving nonlinear equations under the classical Lipschitz conditions for first-order derivatives. To develop a convergence analysis we use the approach of restricted convergence regions in combination to majorizing scalar sequences and our technique of recurrent functions. Finally, the numerical example is given. Full article
(This article belongs to the Special Issue Iterative Methods with Applications in Mathematical Sciences)
Communication
Peculiar Features of Molecular Hydrogen Ions Formed by Proton Collisions with Hydrogen Atoms of the Second Flavor
Foundations 2022, 2(1), 228-233; https://doi.org/10.3390/foundations2010016 - 14 Feb 2022
Viewed by 1182
Abstract
We analyze Molecular Hydrogen Ions (MHIs) formed by collisions of low-energy protons with the Second Flavor of Hydrogen Atoms SFHA, whose existence was previously proven by two kinds of atomic experiments and also evidenced by two kinds of astrophysical observations. We find that [...] Read more.
We analyze Molecular Hydrogen Ions (MHIs) formed by collisions of low-energy protons with the Second Flavor of Hydrogen Atoms SFHA, whose existence was previously proven by two kinds of atomic experiments and also evidenced by two kinds of astrophysical observations. We find that the resulting MHIs would lack a significant number of terms compared to the MHIs formed by collisions of low-energy protons with the usual hydrogen atoms. We show that, in this situation, the radiative transition between the terms of such MHIs of the lowest quantum numbers would be between the terms 5fσ and 4dσ. We calculate the position of the edge of the corresponding molecular band and find it to be at the frequency 14,700 cm−1 or equivalently at the wavelength of 680 nm, which belongs to the visible range. It should be easier to observe this band compared to the spectral bands that are completely beyond the visible range. We emphasize that these results open up another avenue for finding an additional experimental proof of the existence of the SFHA. Namely, if the SFHA is present in gas (in addition to the usual hydrogen atoms), on which a beam of low-energy protons is incident, then the relative intensity of the band, corresponding to the radiative transitions between the terms 5fσ and 4dσ of the MHIs, would be enhanced compared to the absence of the SFHA. Full article
(This article belongs to the Special Issue Advances in Fundamental Physics)
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Article
Binding and Recombination Energies of Quasi-One-Dimensional Excitonic Complexes in Ellipsoidal Quantum Dot
Foundations 2022, 2(1), 219-227; https://doi.org/10.3390/foundations2010015 - 12 Feb 2022
Viewed by 1330
Abstract
In the framework of the effective mass approximation, negative and positive trions, exciton, and biexciton states are investigated in strongly prolate ellipsoidal quantum dots by the variational method. Since the ellipsoidal quantum dot has a prolate character, all excitonic complexes are considered quasi-one-dimensional. [...] Read more.
In the framework of the effective mass approximation, negative and positive trions, exciton, and biexciton states are investigated in strongly prolate ellipsoidal quantum dots by the variational method. Since the ellipsoidal quantum dot has a prolate character, all excitonic complexes are considered quasi-one-dimensional. As in such a system, the analytical solution does not exist for the many-particle problem, it is solved by the variational method. The trial variation functions based on the one-particle wave functions are used to construct the wavefunctions for the excitonic complexes. The energy spectrum, binding, and recombination energies dependent on the geometrical parameters of the ellipsoidal quantum dots are calculated for the excitons, negative and positive trions, and biexcitons. The radiative lifetime of exciton complexes in ellipsoid is estimated. Full article
(This article belongs to the Special Issue Infrared Spectroscopy: Principles and Instrumentation)
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Editorial
Acknowledgment to Reviewers of Foundations in 2021
Foundations 2022, 2(1), 218; https://doi.org/10.3390/foundations2010014 - 11 Feb 2022
Viewed by 873
Abstract
Rigorous peer-reviews are the basis of high-quality academic publishing [...] Full article
Article
On Fractional Lyapunov Functions of Nonlinear Dynamic Systems and Mittag-Leffler Stability Thereof
Foundations 2022, 2(1), 209-217; https://doi.org/10.3390/foundations2010013 - 07 Feb 2022
Viewed by 1181
Abstract
In this paper, fractional Lyapunov functions for epidemic models are introduced and the concept of Mittag-Leffler stability is applied. The global stability of the epidemic model at an equilibrium state is established. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
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Article
Conditions for Scalar and Electromagnetic Wave Pulses to Be “Strange” or Not
Foundations 2022, 2(1), 199-208; https://doi.org/10.3390/foundations2010012 - 07 Feb 2022
Viewed by 1587
Abstract
Vector-valued electromagnetic waves for which the integral of the electric field over time is zero at every location in space were characterized as “usual” by Bessonov several decades ago. Otherwise, they were called “strange”. Recently, Popov and Vinogradov studied conditions leading to usual [...] Read more.
Vector-valued electromagnetic waves for which the integral of the electric field over time is zero at every location in space were characterized as “usual” by Bessonov several decades ago. Otherwise, they were called “strange”. Recently, Popov and Vinogradov studied conditions leading to usual waves using a spectral representation. Their main result is that pulses of finite energy in free space are usual and, consequently, bipolar. However, they do not exclude the possibility of the existence of finite-energy strange pulses, although quite exotic, in a vacuum. Our emphasis in this article is to examine what the relevant necessary and sufficient conditions are for usual and strange waves, particularly for scalar pulses. Illustrative examples are provided, including spherical symmetric collapsing pulses, propagation-invariant, and the so-called almost undistorted spatiotemporally localized waves. Finally, source-generated strange electromagnetic fields are reported. Full article
(This article belongs to the Special Issue Advances in Fundamental Physics)
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Review
The Role of Axions in the Formation of the Photoluminescence Spectrum in Dispersive Media
Foundations 2022, 2(1), 184-198; https://doi.org/10.3390/foundations2010011 - 29 Jan 2022
Cited by 1 | Viewed by 1402
Abstract
In the review, based on the analysis of the results published in the works of domestic and foreign researchers, a variant of an unconventional interpretation of the photoluminescence of dispersive media in the energy range of 0.5–3 eV is proposed. The interpretation meets [...] Read more.
In the review, based on the analysis of the results published in the works of domestic and foreign researchers, a variant of an unconventional interpretation of the photoluminescence of dispersive media in the energy range of 0.5–3 eV is proposed. The interpretation meets the requirements of the energy conservation law for photons and axions participating in the photoluminescence process. The participation of axions in the process is consistent with Primakov’s hypothesis. The role of nonradiative relaxation at the stage of axion decay is noted. The axion lifetimes are estimated for a number of dispersive media. Full article
(This article belongs to the Special Issue Advances in Fundamental Physics)
Article
Analytical Study of a ϕ− Fractional Order Quadratic Functional Integral Equation
Foundations 2022, 2(1), 167-183; https://doi.org/10.3390/foundations2010010 - 25 Jan 2022
Cited by 3 | Viewed by 1391
Abstract
Quadratic integral equations of fractional order have been studied from different views. Here we shall study the existence of continuous solutions of a ϕ fractional-orders quadratic functional integral equation, establish some properties of these solutions and prove the existence of maximal and [...] Read more.
Quadratic integral equations of fractional order have been studied from different views. Here we shall study the existence of continuous solutions of a ϕ fractional-orders quadratic functional integral equation, establish some properties of these solutions and prove the existence of maximal and minimal solutions of that quadratic integral equation. Moreover, we introduce some particular cases to illustrate our results. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
Article
Existence and Uniqueness of Solutions to a Nabla Fractional Difference Equation with Dual Nonlocal Boundary Conditions
Foundations 2022, 2(1), 151-166; https://doi.org/10.3390/foundations2010009 - 21 Jan 2022
Cited by 1 | Viewed by 1626
Abstract
In this paper, we look at the two-point boundary value problem for a finite nabla fractional difference equation with dual non-local boundary conditions. First, we derive the associated Green’s function and some of its properties. Using the Guo–Krasnoselkii fixed point theorem on a [...] Read more.
In this paper, we look at the two-point boundary value problem for a finite nabla fractional difference equation with dual non-local boundary conditions. First, we derive the associated Green’s function and some of its properties. Using the Guo–Krasnoselkii fixed point theorem on a suitable cone and under appropriate conditions on the non-linear part of the difference equation, we establish sufficient requirements for at least one and at least two positive solutions of the boundary value problem. Next, we discuss the existence and uniqueness of solutions to the considered problem. For this purpose, we use Brouwer and Banach fixed point theorem, respectively. Finally, we provide a few examples to illustrate the applicability of established results. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
Article
On the Semi-Local Convergence of a Fifth-Order Convergent Method for Solving Equations
Foundations 2022, 2(1), 140-150; https://doi.org/10.3390/foundations2010008 - 20 Jan 2022
Viewed by 1226
Abstract
We study the semi-local convergence of a three-step Newton-type method for solving nonlinear equations under the classical Lipschitz conditions for first-order derivatives. To develop a convergence analysis, we use the approach of restricted convergence regions in combination with majorizing scalar sequences and our [...] Read more.
We study the semi-local convergence of a three-step Newton-type method for solving nonlinear equations under the classical Lipschitz conditions for first-order derivatives. To develop a convergence analysis, we use the approach of restricted convergence regions in combination with majorizing scalar sequences and our technique of recurrent functions. Finally, a numerical example is given. Full article
(This article belongs to the Special Issue Iterative Methods with Applications in Mathematical Sciences)
Article
Big Rip Scenario in Brans-Dicke Theory
Foundations 2022, 2(1), 128-139; https://doi.org/10.3390/foundations2010007 - 17 Jan 2022
Viewed by 1501
Abstract
In this work, we present a Big Rip scenario within the framework of the generalized Brans-Dicke (GBD) theory. In the GBD theory, we consider an evolving BD parameter along with a self-interacting potential. An anisotropic background is considered to have a more general [...] Read more.
In this work, we present a Big Rip scenario within the framework of the generalized Brans-Dicke (GBD) theory. In the GBD theory, we consider an evolving BD parameter along with a self-interacting potential. An anisotropic background is considered to have a more general view of the cosmic expansion. The GBD theory with a cosmological constant is presented as an effective cosmic fluid within general relativity which favours a phantom field dominated phase. The model parameters are constrained so that the model provides reasonable estimates of the Hubble parameter and other recent observational aspects at the present epoch. The dynamical aspects of the BD parameter and the BD scalar field have been analysed. It is found that the present model witnesses a finite time doomsday at a time of tBR16.14Gyr, and for this scenario, the model requires a large negative value of the Brans-Dicke parameter. Full article
(This article belongs to the Special Issue Advances in Fundamental Physics)
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Article
On the Semi-Local Convergence of a Traub-Type Method for Solving Equations
Foundations 2022, 2(1), 114-127; https://doi.org/10.3390/foundations2010006 - 14 Jan 2022
Cited by 1 | Viewed by 1209
Abstract
The celebrated Traub’s method involving Banach space-defined operators is extended. The main feature in this study involves the determination of a subset of the original domain that also contains the Traub iterates. In the smaller domain, the Lipschitz constants are smaller too. Hence, [...] Read more.
The celebrated Traub’s method involving Banach space-defined operators is extended. The main feature in this study involves the determination of a subset of the original domain that also contains the Traub iterates. In the smaller domain, the Lipschitz constants are smaller too. Hence, a finer analysis is developed without the usage of additional conditions. This methodology applies to other methods. The examples justify the theoretical results. Full article
(This article belongs to the Section Mathematical Sciences)
Article
Relativistic Effects for a Hydrogen Rydberg Atom in a High-Frequency Laser Field: Analytical Results
Foundations 2022, 2(1), 105-113; https://doi.org/10.3390/foundations2010005 - 10 Jan 2022
Viewed by 1379
Abstract
Previously published analytical results for the effects of a high-frequency laser field on hydrogen Rydberg atoms demonstrated that the unperturbed elliptical orbit of the Rydberg electron, generally is engaged simultaneously in the precession of the orbital plane about the direction of the laser [...] Read more.
Previously published analytical results for the effects of a high-frequency laser field on hydrogen Rydberg atoms demonstrated that the unperturbed elliptical orbit of the Rydberg electron, generally is engaged simultaneously in the precession of the orbital plane about the direction of the laser field and in the precession within the orbital plane. These results were obtained while disregarding relativistic effects. In the present paper, we analyze the relativistic effect for hydrogenic Rydberg atoms or ions in a high-frequency linearly- or circularly-polarized laser field, the effect being an additional precession of the electron orbit in its own plane. For the linearly-polarized laser field, the general case, where the electron orbit is not perpendicular to the direction of the laser field, we showed that the precession of the electron orbit within its plane can vanish at some critical polar angle θc of the orbital plane. We calculated analytically the dependence of the critical angle on the angular momentum of the electron and on the parameters of the laser field. Finally, for the particular situation, where the electron orbit is perpendicular to the direction of the laser field, we demonstrated that the relativistic precession and the precession due to the laser field occur in the opposite directions. As a result, the combined effect of these two kinds of the precession is smaller than the absolute value of each of them. We showed that by varying the ratio of the laser field strength F to the square of the laser field frequency ω, one can control the precession frequency of the electron orbit and even make the precession vanish, so that the elliptical orbit of the electron would become stationary. This is a counterintuitive result. Full article
(This article belongs to the Special Issue Advances in Fundamental Physics)
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Article
Divergence in the Relativistic Mean Field Formalism: A Case Study of the Ground State Properties of the Decay Chain of 214,216,218U Isotopes
Foundations 2022, 2(1), 85-104; https://doi.org/10.3390/foundations2010004 - 09 Jan 2022
Cited by 1 | Viewed by 1576
Abstract
A new α-emitting 214U has been recently observed experimentally. This opens the window to theoretically investigate the ground-state properties of the lightest known even–even neutron deficient 214,216,218U isotopes and to examine α-particle clustering around the shell closure. The decay [...] Read more.
A new α-emitting 214U has been recently observed experimentally. This opens the window to theoretically investigate the ground-state properties of the lightest known even–even neutron deficient 214,216,218U isotopes and to examine α-particle clustering around the shell closure. The decay half-lives are calculated within the preformed cluster-decay model (PCM). To obtain the α-daughter interaction potential, the RMF densities are folded with the newly developed R3Y and the well-known M3Y NN potentials for comparison. The alpha preformation probability (Pα) is calculated from the analytic formula of Deng and Zhang. The WKB approximation is employed for the calculation of the transmission probability. The individual binding energies (BE) for the participating nuclei are estimated from the relativistic mean-field (RMF) formalism and those from the finite range droplet model (FRDM) as well as WS3 mass tables. In addition to Z=84, the so-called abnormal enhancement region, i.e., 84Z90 and N<126, is normalised by an appropriately fitted neck-parameter ΔR. On the other hand, the discrepancy sets in due to the shell effect at (and around) the proton magic number Z=82 and 84, and thus a higher scaling factor ranging from 108105 is required. Additionally, in contrast with the experimental binding energy data, large deviations of about 5–10 MeV are evident in the RMF formalism despite the use of different parameter sets. An accurate prediction of α-decay half-lives requires a Q-value that is in proximity with the experimental data. In addition, other microscopic frameworks besides RMF could be more reliable for the mass region under study. α-particle clustering is largely influenced by the shell effect. Full article
(This article belongs to the Special Issue Advances in Fundamental Physics)
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Article
On the Topological Structure of Nonlocal Continuum Field Theories
Foundations 2022, 2(1), 20-84; https://doi.org/10.3390/foundations2010003 - 31 Dec 2021
Cited by 3 | Viewed by 1635
Abstract
An alternative to conventional spacetime is proposed and rigorously formulated for nonlocal continuum field theories through the deployment of a fiber bundle-based superspace extension method. We develop, in increasing complexity, the concept of nonlocality starting from general considerations, going through spatial dispersion, and [...] Read more.
An alternative to conventional spacetime is proposed and rigorously formulated for nonlocal continuum field theories through the deployment of a fiber bundle-based superspace extension method. We develop, in increasing complexity, the concept of nonlocality starting from general considerations, going through spatial dispersion, and ending up with a broad formulation that unveils the link between general topology and nonlocality in generic material media. It is shown that nonlocality naturally leads to a Banach (vector) bundle structure serving as an enlarged space (superspace) inside which physical processes, such as the electromagnetic ones, take place. The added structures, essentially fibered spaces, model the topological microdomains of physics-based nonlocality and provide a fine-grained geometrical picture of field–matter interactions in nonlocal metamaterials. We utilize standard techniques in the theory of smooth manifolds to construct the Banach bundle structure by paying careful attention to the relevant physics. The electromagnetic response tensor is then reformulated as a superspace bundle homomorphism and the various tools needed to proceed from the local topology of microdomains to global domains are developed. For concreteness and simplicity, our presentations of both the fundamental theory and the examples given to illustrate the mathematics all emphasize the case of electromagnetic field theory, but the superspace formalism developed here is quite general and can be easily extended to other types of nonlocal continuum field theories. An application to fundamental theory is given, which consists of utilizing the proposed superspace theory of nonlocal metamaterials in order to explain why nonlocal electromagnetic materials often require additional boundary conditions or extra input from microscopic theory relative to local electromagnetism, where in the latter case such extra input is not needed. Real-life case studies quantitatively illustrating the microdomain structure in nonlocal semiconductors are provided. Moreover, in a series of connected appendices, we outline a new broad view of the emerging field of nonlocal electromagnetism in material domains, which, together with the main superspace formalism introduced in the main text, may be considered a new unified general introduction to the physics and methods of nonlocal metamaterials. Full article
(This article belongs to the Special Issue Advances in Fundamental Physics)
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Article
Analytical and Qualitative Study of Some Families of FODEs via Differential Transform Method
Foundations 2022, 2(1), 6-19; https://doi.org/10.3390/foundations2010002 - 28 Dec 2021
Viewed by 1125
Abstract
This current work is devoted to develop qualitative theory of existence of solution to some families of fractional order differential equations (FODEs). For this purposes we utilize fixed point theory due to Banach and Schauder. Further using differential transform method (DTM), we also [...] Read more.
This current work is devoted to develop qualitative theory of existence of solution to some families of fractional order differential equations (FODEs). For this purposes we utilize fixed point theory due to Banach and Schauder. Further using differential transform method (DTM), we also compute analytical or semi-analytical results to the proposed problems. Also by some proper examples we demonstrate the results. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
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Communication
Possible Observational Evidence for the Existence of a Parallel Universe
Foundations 2022, 2(1), 1-5; https://doi.org/10.3390/foundations2010001 - 23 Dec 2021
Cited by 1 | Viewed by 2329
Abstract
Many totally different kinds of astrophysical observations demonstrated that, in our universe, there exists a preferred direction. Specifically, from observations in a wide range of frequencies, the alignment of various preferred directions in different data sets was found. Moreover, the observed Cosmic Microwave [...] Read more.
Many totally different kinds of astrophysical observations demonstrated that, in our universe, there exists a preferred direction. Specifically, from observations in a wide range of frequencies, the alignment of various preferred directions in different data sets was found. Moreover, the observed Cosmic Microwave Background (CMB) quadrupole, CMB octopole, radio and optical polarizations from distant sources also indicate the same preferred direction. While this hints at a gravitational pull from the “outside”, the observational data from the Plank satellite showed that the bulk flow velocity was relatively small: much smaller than was initially thought. In the present paper we propose a configuration where two three-dimensional universes (one of which is ours) are embedded in a four-dimensional space and rotate about their barycenter in such a way that the centrifugal force nearly (but not exactly) compensates their mutual gravitational pull. This would explain not only the existence of a preferred direction for each of the three-dimensional universes (the direction to the other universe), but also the fact that the bulk flow velocity, observed in our universe, is relatively small. We point out that this configuration could also explain the perplexing features of the Unidentified Aerial Phenomena (UAP), previously called Unidentified Flying Objects (UFOs), recorded by various detection systems—the features presented in the latest official report by the US Office of the Director of National Intelligence. Thus, the proposed configuration of the two rotating, parallel three-dimensional universes seems to explain both the variety of astrophysical observations and (perhaps) the observed features of the UAP. Full article
(This article belongs to the Special Issue Advances in Fundamental Physics)
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