Foundations, Volume 4, Issue 4 (December 2024) – 17 articles

This work explores the hyperbolic cosine and hyperbolic secant functions within the framework of the maximum entropy principle, deriving these probability distribution functions from first principles. The resulting maximum entropy solutions are applied to various physical systems, including the repulsive oscillator and solitary wave solutions of the advection equation, using the method of moments. Additionally, a different moment analysis using experimental and theoretical inputs is employed to address non-linear systems described by the non-linear Schrödinger equation, non-linear diffusion equation, and Korteweg–de Vries equation, demonstrating the versatility of this approach. These findings demonstrate the broad applicability of maximum entropy methods in solving different differential equations, with potential implications for future research in non-linear dynamics and transport physics. Full article

The rising demand for both water and energy has intensified the urgency of addressing the water–energy nexus. Energy is required for water treatment and distribution, and energy production processes require water. The increasing demand for energy requires substantial amounts of water, primarily for cooling. The emergence of new persistent contaminants has necessitated the use of advanced, energy-intensive water treatment methods. Coupled with the energy demands of water distribution, this has significantly strained the already limited energy resources. Regrettably, no straightforward, universal model exists for estimating water usage and energy consumption in power and water treatment plants, respectively. Current approaches rely on data from direct surveys of plant operators, which are often unreliable and incomplete. This has significantly undermined the efficiency of the plants as these surveys often miss out on complex interactions, lack robust predictive power and fail to account for dynamic temporal changes. The study thus aims to evaluate the potential of mathematical modeling and simulation in the water–energy nexus. It formulates a mathematical framework and subsequent simulation in Java programming to estimate the water use in hydroelectric power and geothermal energy, the energy consumption of the advanced water treatment processes focusing on advanced oxidation processes and membrane separation processes and energy demands of water distribution. The importance of mathematical modeling and simulation in the water–energy nexus has been extensively discussed. The paper then addresses the challenges and prospects and provides a way forward. The findings of this study strongly demonstrate the effectiveness of mathematical modeling and simulation in navigating the complexities of the water–energy nexus. Full article

The framework of this paper is the presentation of a case study in which university students are required to extend a particular problem of division of polynomials in one variable over the field of real numbers (as generalizing action) clearly influenced by prior strategies (as reflection generalization). Specifically, the objective of this paper is to present a methodology for generalizing the classical Remainder Theorem to the case in which the divisor is a product of binomials ${(x-a_1)}^{n_1}{(x-a_2)}^{n_2}\cdots {(x-a_k)}^{n_k}$, where $a_1,a_2,\cdots ,a_k\in \mathbb{R}$ and $n_1,n_2,\cdots ,n_k\in \mathbb{N}$. A first approach to this issue is the Taylor expansion of the dividend $P\left(x\right)$ at a point a, which clearly shows the quotient and the remainder of the division of $P\left(x\right)$ by ${(x-a)}^k$, where the degree of $P\left(x\right)$, say n, must be greater than or equal to k. The methodology used in this paper is the proof by induction which allows to obtain recurrence relations different from those obtained by other scholars dealing with the generalization of the classical Remainder Theorem. Full article

The empirical wavelet transform is a fully adaptive time-scale representation that has been widely used in the last decade. Inspired by the empirical mode decomposition, it consists of filter banks based on harmonic mode supports. Recently, it has been generalized to build the filter banks from any generating function using mappings. In practice, the harmonic mode supports can have a low-constrained shape in 2D, leading to numerical difficulties to estimate mappings adapted to the construction of empirical wavelet filters. This work aims to propose an efficient numerical scheme to compute empirical wavelet coefficients using the demons registration algorithm. Results show that the proposed approach is robust, accurate, and continuous wavelet filters permitting reconstruction with a low signal-to-noise ratio. An application for texture segmentation of scanning tunneling microscope images is also presented. Full article

The Wilson–Cowan model has been widely applied for the simulation of electroencephalography (EEG) waves associated with neural activities in the brain. The Runge–Kutta (RK) method is commonly used to numerically solve the Wilson–Cowan equations. In this paper, we focus on enhancing the accuracy of the numerical method by proposing a strategy to construct a class of fourth-order RK methods using a generalized iterated Crank–Nicolson procedure, where the RK coefficients depend on a free parameter $c_2$. When $c_2$ is set to 0.5, our method becomes a special case of the classical fourth-order RK method. We apply the proposed methods to solve the Wilson–Cowan equations with two and three neuron populations, modeling EEG epileptic dynamics. Our simulations demonstrate that when $c_2$ is set to 0.4, the proposed RK4-04 method yields smaller errors compared to those obtained using the classical fourth-order RK method. This is particularly visible when the spectral radius of the connection matrix or the excitation-inhibition coupling coefficient is relatively large. Full article

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’s 1964 paper (parameter independence) is insufficient for the derivation of determinism. Attention is then turned to critically assess Bell’s appeal to the Einstein–Rosen–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 “counterfactual” reality criterion enables a rigorous derivation of both determinism and parameter independence, and in this sense justifies Bell’s claim. Conversely, however, it is noted that whereas the EPR argument requires a weaker “measurement choice” assumption than Bell’s demonstration, it nevertheless leads to a similar incompatibility with quantum predictions rather than quantum incompleteness. Full article

This paper presents a comprehensive analytical–numerical algorithm constructed for proprotor performance evaluation, focusing on accommodating large inflow angles. The algorithm’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 −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. Full article

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–de Vries equation (KdV), the Kuramoto–Sivashinsky equation (KS), the generalized Korteweg–de Vries–Kuramoto–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 $\mathcal{N}\left(u_x\right)$, where $u_x$ denotes the derivative of u with respect to x, focusing in particular on the Kuramoto–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–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–Sivashinsky equation in ${\mathbb{R}}^n$. 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. Full article

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 $T>0$ and $dS<0$ for $T<0$. 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. Full article

We consider spontaneous quark transitions between the ${\mathsf{\Lambda }}^0$ baryon and its resonant states, and (anti)quark transitions between the neutral kaon K⁰ and the two heavy ${\mathsf{\eta }}_{\mathrm{q}}$-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 ${\mathsf{\Lambda }}_{\mathrm{c}}^{+}$-baryon. We find that the flips $\mathrm{s}\to \mathrm{c}$ and $\overline{\mathrm{s}}\to \overline{\mathrm{c}}$ 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 $\overline{\mathrm{d}}$-ground. The wider quark range stems from the flip $\mathrm{s}\to \mathrm{b}$, which costs 283 MeV more (or $3\times$ more) than the corresponding antiquark flip $\overline{\mathrm{s}}\to \overline{\mathrm{b}}$. At the same time, transitions from the respective ground states to the s and $\overline{\mathrm{s}}$ states (or the c and $\overline{\mathrm{c}}$ 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 $\mathsf{\Lambda }$-baryons and B-mesons. Full article

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. Full article

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’ structure. We start from exact solutions of Dirac’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 ${}^1\mathrm{H}$, ${}^2\mathrm{H}$, ${}^3\mathrm{H}$, and ${}^3\mathrm{He}^{+}$, 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’s equation within the nuclear core. Larger differences between different nuclear structure models are found in the case of the anisotropic $2p_{3/2}$ orbitals of hydrogen, rendering these excited states as promising reference systems for exploring the proton structure. Full article

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–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–Stokes–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. Full article

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. Full article