Foundations doi: 10.3390/foundations6020020

Authors: Maria Emília Bigotte de Almeida João Ricardo Branco Carla Fidalgo Luís Margalho

Students entering engineering programs often exhibit insufficient mathematics knowledge and considerable variability in prior training, which can create learning gaps and challenges for higher education integration. This study aims to characterize students’ mathematics proficiency at the Coimbra Institute of Engineering and to develop strategies to address these gaps. A diagnostic test was designed based on the Portuguese primary and secondary education syllabus and the guidelines of the European Society for Engineering Education. Data were collected from students enrolling in engineering degrees between the 2013/14 and 2021/22 academic years. Based on the diagnostic results, a targeted intervention was implemented to motivate students and enhance their learning in mathematics. This intervention includes complementary teaching methodologies applied to Differential and Integral Calculus, a mandatory first-year course across all engineering programs. The analysis demonstrates that the combined approach of diagnostic assessment and targeted support improves student engagement and addresses disparities in prior knowledge. This study contributes to the development of evidence-based strategies that support equitable learning opportunities in engineering education and offers a model for integrating diagnostic assessment with active learning practices in foundational STEM courses.

Foundations doi: 10.3390/foundations6020019

Authors: Stanislav Srednyak

In this paper, we discuss extensions of the canonical quantization procedure in quantum field theories. We focus specifically on S-matrix representation as a T-exponent. This extension involves flat bundles on certain infinite dimensional functional manifolds of local time. The motivating problem is first principles treatment of bound states in quantum chromodynamics as well as precision physics of the hydrogen atom and the muonium. Our main results include systematic treatment of flat bundles in an infinite dimensional setting, generalization of Hamiltonian evolution and functional renormalization group evolution equations in quantum field theories. We discuss several results from finite dimensional theory that have analogies in the functional setting. This includes construction of moduli space of flat connections and isomonodromic deformations. One of the outcomes of our analysis is a construction of a rich family of functional flat bundles with rational connections. This class of connections exhibits a rich set of mathematical properties. In particular, we construct examples of the fundamental groups of spaces which have a definable continuum of generators. Physical states correspond to points in the moduli space of bundles on these spaces. On the physics side of things, we conclude that spacetime notions, such as spaces of particle configurations, emerge effectively as spectral sets of functional differential operators.

Foundations doi: 10.3390/foundations6020018

Authors: Tim Pollmann Jochen Staudacher

The Shapley value is the predominant point-valued solution concept in cooperative game theory and has recently become a foundational method in interpretable machine learning. In this domain, a prevailing strategy for circumventing the computational intractability of exact Shapley values is to approximate them via a weighted least squares optimization framework. In this paper, we investigate an existing algorithmic framework for weighted least squares Shapley approximation, assessing its feasibility for feature attribution. Methodologically, we conduct a theoretical variance analysis within a Monte Carlo sampling framework, investigate an approach for sample reuse across strata, and establish a relation to Unbiased KernelSHAP. Our analysis reveals three main findings: (i) a structural equivalence between least squares sampling and Unbiased KernelSHAP; (ii) the non-zero covariance between sampled coalitions introduced by reusing samples across strata in one of the existing least squares-based approaches; and (iii) the absence of a universally optimal sampling strategy across tasks. We validate these results empirically on several cooperative games and practical machine learning problems.

Foundations doi: 10.3390/foundations6020017

Authors: Sebastian Pardo-Guerra Anil Thapa Megan Simons Jonathan Washburn

We develop an information-theoretic, cost-first framework for discrete dynamics in which the primitive operation is ratio-based comparison. Given two quantities compared via their ratio x=a/b, we assign a cost F(x) measuring deviation from equilibrium (x=1). Adopting a reciprocal d’Alembert composition law motivated by coherent chaining, together with quadratic calibration at unity, uniquely determines a reciprocal comparison cost J(x)=12x+x−1−1. Taking J as input, we model recognition events as deterministic updates on directed graphs recorded in a minimal ledger. Minimality (no intra-tick ordering metadata) together with non-commutativity of events implies atomic ticks: at most one event per tick. With conservation, pairwise locality, and quantization in δZ, each event is recorded as a balanced double-entry posting. For graphs with cycles, assuming time-aggregated cycle closure over a finite clearing horizon, we show that cleared cycle closure is equivalent to path-independence and that the cumulative flow admits a scalar potential on each connected component (unique up to additive constant) via a discrete Poincaré lemma. On hypercube graphs Qd, atomic single-edge updates impose a 2d-tick minimal period for timestamp-unique coverage, realized by cyclic Gray codes (explicitly for d=3). The framework links ratio-based cost functions, conservative graph flows, and discrete potential theory through explicitly stated axioms and structural assumptions.

Foundations doi: 10.3390/foundations6020016

Authors: José A. Rodrigues

Time–space fractional diffusion equations are widely used to model anomalous transport in heterogeneous biological tissues, where memory effects, spatial nonlocality, and coefficient variability are intrinsically coupled. However, existing numerical approaches typically treat these aspects in isolation, and a fully discrete framework that simultaneously accounts for heterogeneity, long-memory effects, and computational efficiency remains lacking. In this work, a fully discrete numerical method is developed and analyzed. The method integrates heterogeneous diffusion coefficients and memory-efficient temporal discretization within a unified variational framework. It combines a finite element approximation of a spectral fractional elliptic operator with an implicit L1 discretization of the Caputo derivative enhanced by a sum-of-exponentials approximation of the memory kernel. Unconditional stability, preservation of a discrete energy structure, and a fully discrete error estimate are established, explicitly separating temporal, spatial, and kernel approximation errors. The proposed approach reduces memory complexity from O(N) to O(logN) without compromising accuracy. Numerical experiments confirm the theoretical convergence rates, demonstrate stable behavior across all tested configurations, and illustrate the impact of heterogeneous coefficients on anomalous transport dynamics.

Foundations doi: 10.3390/foundations6020015

Authors: Ioannis K. Argyros Jinny Ann John Samundra Regmi

The convergence order of higher-order iterative methods for solving systems of nonlinear equations was analyzed using Taylor series expansion, which typically requires the computation of higher-order derivatives not inherently part of the method. This dependency limits the method’s applicability and increases the computational cost. The distinctiveness of our work lies in the development of improved convergence theorems that rely solely on first-order derivatives. The proposed approach offers a stronger framework than existing methods by incorporating details about the convergence region’s radius and providing precise error estimates. Furthermore, we explore semi-local convergence, which holds greater significance as it allows the identification of the specific domain where the iterative sequence remains valid. The theoretical findings are substantiated through suitable numerical illustrations.

Foundations doi: 10.3390/foundations6020014

Authors: Antonio E. Bargellini Daniele Ritelli

We present a proof, using elementary methods, of the Euler reflection formula for the Gamma function, based on an integral computed by Laplace and on the Euler–Gauss infinite product representation of Gamma. This way, we reverse the classical path, and, using the reflection formula as a starting point, we obtain the representation of the sine as an infinite product and that of the cotangent in partial fractions, which, as is known, allows the explicit calculation of the zeta function with an even argument: all this without resorting to complex analysis or the Herglotz trick. We can present a teaching proposal that illustrates the complete proof of this fundamental formula using undergraduate-level mathematical analysis tools, such as the derivation of parametric integrals, the second Mean Value Theorem for Integrals (Bonnet formula), and the convergence criterion for Dirichlet oscillatory integrals.

Foundations doi: 10.3390/foundations6020013

Authors: Angelo Plastino

We show that thermodynamic consistency in systems with finite excluded volume implies compact support of the grand canonical particle-number distribution. Understanding whether fundamental bounds on information and matter content can arise purely from statistical-mechanical principles—independent of gravitational dynamics—is of central interest in thermodynamics, information theory, and cosmology. For any nonzero excluded volume parameter b, the partition function vanishes identically beyond Nmax=V/b, enforcing a strict upper bound on admissible macrostates. We demonstrate that this compactness induces bounded particle-number fluctuations and finite Fisher information with respect to the chemical potential, thereby rendering the associated statistical manifold effectively finite-dimensional. This informational compactness provides a structural mechanism limiting distinguishability of macrostates independently of gravitational considerations. We argue that such thermodynamically enforced bounds are compatible with entropy bounds and holographic scaling principles, suggesting that informational finiteness may arise from statistical-mechanical consistency alone. Cosmological implications are discussed cautiously: infinite matter content at fixed volume is incompatible with compact support induced by finite excluded volume. Accordingly, the Fisher metric and associated thermodynamic lengths remain bounded when particle-number fluctuations are restricted by excluded-volume constraints. These results show that excluded-volume constraints induce a natural information-geometric compactness of the thermodynamic manifold, providing a general mechanism by which statistical distinguishability and curvature remain finite in finite-occupancy systems.

Foundations doi: 10.3390/foundations6010012

Authors: Furkan Erkan Nuket Aykut Hamal Sotiris K. Ntouyas Bashir Ahmad

This work focuses on the analysis of a sequential fractional boundary value problem involving coupled Erdélyi–Kober and Caputo fractional differential operators, together with nonlocal boundary conditions of fractional type. The well-posedness of the problem is addressed by deriving conditions that ensure the existence and uniqueness of solutions. Uniqueness is obtained through an application of Banach’s contraction principle, whereas existence is established by employing Krasnosel’skiĭ’s fixed point approach and the nonlinear alternative of Leray–Schauder. Several numerical examples are presented to demonstrate and support the theoretical findings.

Foundations doi: 10.3390/foundations6010011

Authors: Olga V. Soledad-Flores Sonia J. Bailón-Ruiz

Carbon dots (C-Dots) have attracted significant interest due to their strong photoluminescence, aqueous stability, and tunable surface chemistry; however, their environmental safety remains incompletely understood. In this work, C-Dots were synthesized via a rapid microwave-assisted method using two different carbon precursors, D-glucose and ascorbic acid, with ethylenediamine as a passivating agent. The resulting nanoparticles exhibited predominantly amorphous structures with sizes below 10 nm, characteristic absorption bands at ~280–330 nm, and blue photoluminescence centered at ~450 nm. Acute toxicity was evaluated using Brine shrimp at concentrations ranging from 10 to 2000 ppm after 24 and 48 h of exposure. C-Dots synthesized from ascorbic acid showed significant toxicity at 2000 ppm, inducing higher mortality rates after 24 h, whereas D-glucose-derived C-Dots exhibited minimal toxic effects under the same conditions. These findings demonstrate that carbon precursor selection plays a critical role in determining the environmental toxicity of C-Dots and highlight the importance of precursor-dependent design strategies to minimize potential ecological risks associated with carbon-based nanomaterials.

Foundations doi: 10.3390/foundations6010010

Authors: Protopapas Eleftherios

Aquatic pollution threatens biodiversity, disrupts ecological balance, and poses risks to communities dependent on freshwater resources. Aquaculture ponds are especially susceptible, as contaminants directly influence both ecosystem stability and the safety of fish for human consumption. With the rapid growth of pond-based aquaculture, accurate modeling of pollutant dynamics is essential. This study analyzes pollution in a system of n interconnected ponds, assuming a clean water source, constant volume, and steady pollutant inflow and outflow. A previous model based on ordinary differential equations is solved using matrices, eigenvalues, eigenvectors, and generalized eigenvectors. A generalized fractional model is then developed employing the Caputo–Liouville derivative. Unlike classical models, fractional models account for memory effects and anomalous diffusion, providing a more realistic description of pollutant behavior. Analytical solutions are derived to track pollutant variation across ponds, and a comparison of the two formulations is presented. The results enhance understanding of pollution transport in aquaculture systems and offer insights for sustainable water quality management in fish farming.

Foundations doi: 10.3390/foundations6010009

Authors: Juan Núñez Valdés Fernando de Pablos Pons Antonio Ramos Carrillo

Figure 8 and the corresponding reference in the original publication [...]

Foundations doi: 10.3390/foundations6010008

Authors: Georgios Angelidis

Global value chains (GVCs) have evolved into highly interconnected and geographically fragmented production networks, increasing exposure to systemic disruptions and revealing the limitations of static input–output and conventional network approaches. This study develops a unified analytical framework for modeling the structure, dynamics, and resilience of GVCs by integrating input–output economics with network theory, control theory, optimal transport, information theory, and cooperative game theory. The framework represents GVCs as time-varying, multi-level networks and formalizes shock propagation through stochastic normalization and state-space dynamics. Entropy-regularized optimal transport is employed to model friction-dependent substitution and supply chain reconfiguration, while Koopman operator methods approximate nonlinear adjustment dynamics. Cooperative flow-based indices are introduced to assess systemic importance and bargaining power. The analysis produces a coherent set of structural and dynamic indicators capturing vulnerability, adaptability, and controllability across country–sector nodes. Overall, the framework provides an empirically applicable toolkit for diagnosing structural fragilities, comparing resilience across economies, and supporting scenario-based evaluation of industrial and trade policies in complex global production networks.

Foundations doi: 10.3390/foundations6010007

Authors: Hans-Otto Carmesin

Einstein derived the expansion of space ever since the Big Bang started and introduced the possible cosmological constant Λ. The expansion of space and the present-day expansion rate H0, the Hubble constant, has been discovered by Hubble. Perlmutter discovered the positive value of Λ, and Zeldovich showed that Λ corresponds to the energy density uDE of space. Lamb and Retherford as well as Casimir provided evidence for the idea that uDE might be based on quanta, and Riess et al. provided evidence that H0 is an idealization. In this paper, using the hypothetico-deductive method with very founded hypotheses, these two pieces of evidence are confirmed in a very founded and precise manner. Thereby, neither a fit is executed, nor a postulate, nor an unfounded hypothesis is proposed.

Foundations doi: 10.3390/foundations6010006

Authors: Kishor Chandramohan Jagadeeswari Murugan Thiruvenkadam Srinivasan In-Ho Ra

A novel approach for analyzing the structural integrity and operational vulnerability of complex networks using intuitionistic fuzzy graphs has been modeled. While traditional fuzzy graph metrics focus primarily on existence, they fail to capture the holistic systemic impact of failures. To overcome this limitation, a scalar-based measure of nodal importance that integrates both existence (membership degree) and non-existence (non-membership degree) values of incident edges into a single critical metric has been developed. The proposed indices demonstrate enhanced sensitivity to network perturbations compared to conventional degree centrality measures, capturing latent vulnerabilities in critical infrastructure topologies. Based on this, two indices are proposed: Intuitionistic Fuzzy Degree Index and Intuitionistic Edge Interaction Index. These indices quantify the total system activity, stress dispersion, overall network cohesiveness, and potential for cascading failure propagation. When applied to synthetic core-periphery networks, the proposed indices identified critical nodes with superior discrimination capability compared to existing fuzzy graph metrics, revealing that removal of identified nodes results in system-wide connectivity degradation observable through both membership and non-membership approximations. This methodology was applied to a core-periphery communication network to analyze the systemic consequences of node removal. Experimental validation on networks of varying sizes demonstrates that the Intuitionistic Edge Interaction Index achieves robust node criticality ranking across heterogeneous network topologies with improved predictive accuracy for cascade initiation points. This work provides network analysts and engineers a quantitative tool to precisely assess criticality and inform targeted resilience strategies in uncertain, high-risk environments.

Foundations doi: 10.3390/foundations6010005

Authors: Flavia Pennini Angelo Plastino

This study examines the time evolution of structural and informational quantifiers in a damped Rabi oscillator, specifically focusing on fidelity, entropy, disequilibrium, and Fisher information. We observe that all four measures exhibit damped oscillatory behavior as the system approaches its steady state. However, the final asymptotic behavior is striking: while fidelity and disequilibrium indicate a residual, non-zero final state, and entropy quantifies the thermodynamic disorder, Fisher information uniquely vanishes. This vanishing implies a complete loss of dynamical information—the ability to infer the system’s past evolution from its current state—even in the absence of complete thermodynamic disorder. Our findings introduce a new phenomenon where a system can be “informationally silent”, meaning it becomes structurally ordered yet loses all inferential sensitivity to its own history, a detail that traditional entropy measures do not fully capture. This work highlights a critical distinction between thermodynamic disorder (entropy) and inferential sensitivity (Fisher information) in the context of open quantum systems.

Foundations doi: 10.3390/foundations6010004

Authors: Călin-Ioan Gheorghiu Eduard S. Grigoriciuc

The well-known shooting algorithm has produced important results in solving various linear as well as nonlinear BVPs, defined on unbounded intervals, but has become obsolete. The main difficulty lies in the numerical handling of the domain’s infiniteness. This paper presents a three-step strategy that significantly improves the traditional truncation algorithm. It consists of Chebyshev collocation, implemented as Chebfun, in conjunction with rational AAA interpolation and analytic continuation. Furthermore, and more importantly, this approach enables us to provide a thorough analysis of both possible errors in dealing with and the hidden singularities of some BVPs of real interest. A singular second-order eigenvalue problem and a fourth-order nonlinear degenerate parabolic equation, all defined on the real axis, are considered. For the latter, Chebfun provides properties-preserving solutions. Travelling wave solutions are also studied. They are highly nonlinear BVPs. The problem arises from the analysis of thin viscous film flows down an inclined plane under the competing stress due to the surface tension gradients and gravity, a long-standing concern of ours. By extending the solutions to these problems in the complex plane, we observe that the complex poles do not influence their behaviour. On the other hand, the real ones involve singularities and indicate how long solutions can be extended through continuity.

Foundations doi: 10.3390/foundations6010003

Authors: Domenico Cantone Andrea De Domenico Pietro Maugeri Eugenio G. Omodeo

As a contribution to automated set-theoretic inferencing, a translation is proposed of conjunctions of literals of the forms x=y∖z, x≠y∖z, and z=x, where x,y,z stand for variables ranging over the von Neumann universe of sets, into quantifier-free Boolean formulae of a rather simple conjunctive normal form. The formulae in the target language involve variables ranging over a Boolean ring of sets, along with a difference operator and relators designating equality, non-disjointness, and inclusion. Moreover, the result of each translation is a conjunction of literals of the forms x=y∖z and x≠y∖z and of implications whose antecedents are isolated literals and whose consequents are either inclusions (strict or non-strict) between variables, or equalities between variables. Besides reflecting a simple and natural semantics, which ensures satisfiability preservation, the proposed translation has quadratic algorithmic time complexity and bridges two languages, both of which are known to have an NP-complete satisfiability problem.

Foundations doi: 10.3390/foundations6010002

Authors: Glykeria Kyrou Vasileios Charilogis Ioannis G. Tsoulos

Global optimization represents a fundamental challenge in computer science and engineering, as it aims to identify high-quality solutions to problems spanning from moderate to extremely high dimensionality. The Differential Evolution (DE) algorithm is a population-based algorithm like Genetic Algorithms (GAs) and uses similar operators such as crossover, mutation and selection. The proposed method introduces a set of methodological enhancements designed to increase both the robustness and the computational efficiency of the classical DE framework. Specifically, an adaptive termination criterion is incorporated, enabling early stopping based on statistical measures of convergence and population stagnation. Furthermore, a population sampling strategy based on k-means clustering is employed to enhance exploration and improve the redistribution of individuals in high-dimensional search spaces. This mechanism enables structured population renewal and effectively mitigates premature convergence. The enhanced algorithm was evaluated on standard large-scale numerical optimization benchmarks and compared with established global optimization methods. The experimental results indicate substantial improvements in convergence speed, scalability and solution stability.

Foundations doi: 10.3390/foundations6010001

Authors: Hyoung-In Lee Sang-Hyeon Kim Tae-Yeon Kim Hee-Jeong Moon

The structural vibration of industrial droplet dispensers can be modeled by telegraph-like equations to a good approximation. We reinterpret the telegraph equation from the standpoint of an electric–circuit system consisting of an inductor and a resistor, which is in interaction with an environment, say, a substrate. This interaction takes place through a capacitor and a shunt resistor. Such interactions serve as leakage. We have performed an analytical investigation of the frequency dispersion of telegraph equations over an unbounded one-dimensional domain. By varying newly identified key parameters, we have not only recovered the well-known characteristics but also discovered crossover phenomena regarding phase and group velocities. We have examined frequency responses of the electric circuit underlying telegraph equations, thereby confirming the role as low-pass filters. By identifying a set of physically meaningful reduced cases, we have laid the foundations on which we could further explore wave propagations over a finite domain with appropriate side conditions.

Foundations doi: 10.3390/foundations5040037

Authors: Furkan Erkan Nuket Aykut Hamal Sotiris K. Ntouyas Bashir Ahmad

In this paper, we investigate a new class of nonlinear fractional boundary value problems (BVPs) involving (k,ψ)-Caputo fractional derivative operators subject to multipoint closed boundary conditions. Such a formulation of boundary data generalizes classical closure constraints in terms of nonlocal dependence of the unknown function at several interior points, giving rise to a flexible mechanism for describing physical and engineering phenomena governed by nonlocal and memory effects. The proposed problem is first transformed into an equivalent fixed-point formulation, enabling the application of standard analytical tools. Results concerning the existence and uniqueness of solutions to the problem are obtained through the application of fixed-point principles, specifically those of Banach, Krasnosel’skiĭ, and the Leray–Schauder nonlinear alternative. The obtained results extend and generalize several known findings. Illustrative examples are presented to demonstrate the applicability of the theoretical findings. Moreover, the introduction incorporates a succinct review of boundary value problems associated with fractional differential equations and inclusions subject to closed boundary conditions.

Foundations doi: 10.3390/foundations5040036

Authors: Alexander Robitzsch

The root mean square deviation (RMSD) is a widely used item fit statistic in item response models. However, the sample RMSD is known to exhibit positive bias in small samples. To address this, seven alternative bias-corrected RMSD estimators are proposed and evaluated in a simulation study involving items with uniform differential item functioning (DIF). The results demonstrate that the proposed estimators effectively reduce the bias of the original RMSD statistic. Their performance is compared, and the most favorable estimators are highlighted for empirical research. Finally, the application of the various RMSD statistics is illustrated using PISA 2006 reading data.

Foundations doi: 10.3390/foundations5040035

Authors: Ty Roachford Warren Mansell Rodrigo Pena

Perceptual Control Theory (PCT) and the Free Energy Principle (FEP) are two foundational, principle-based frameworks originally developed to explain brain function. However, since their initial proposals, both frameworks have been generalized to account for the behavior of living systems more broadly. Despite their conceptual overlap and practical successes, a mathematical comparison of the two frameworks has yet to be undertaken. In this article, we briefly introduce and compare the philosophical foundations underlying PCT and FEP. We then introduce and compare their experimental and mathematical foundations concretely in the context of bacterial chemotaxis. With these foundations in place, we can use tools from category theory to argue that PCT can be formally understood as a subset of the FEP framework; however, it is worth noting that the mathematical machinery unique to FEP is not required to successfully model bacterial chemotaxis. Finally, we conclude with a proposal for a mathematical synthesis where each framework plays an orthogonal yet complementary role.

Foundations doi: 10.3390/foundations5040034

Authors: Edward Bormashenko

A probabilistic version of geometry is introduced. The fifth postulate of Euclid (Playfair’s axiom) is adopted in the following probabilistic form: consider a line and a point not on the line—there is exactly one line through the point with probability P, where 0≤P≤1. Playfair’s axiom is logically independent of the rest of the Hilbert system of axioms of the Euclidian geometry. Thus, the probabilistic version of the Playfair axiom may be combined with other Hilbert axioms.  P=1 corresponds to the standard Euclidean geometry; P=0 corresponds to the elliptic- and hyperbolic-like geometries. 0<P<1 corresponds to the introduced probabilistic geometry. Parallel constructions in this case are Bernoulli trials. Theorems of the probabilistic geometry are discussed. Given a triangle and a line drawn from a vertex parallel to the opposite side, the event that this line is actually parallel occurs with probability P. Otherwise, the line may intersect the side or diverge. Parallelism is not transitive in the probabilistic geometry. Probabilistic geometry occurs on the surface with a stochastically variable Gaussian curvature. Alternative geometries adopting various versions of the probabilistic Playfair axiom are introduced. Probabilistic non-Archimedean geometry is addressed. Applications of the probabilistic geometry are discussed.

Foundations doi: 10.3390/foundations5040033

Authors: Adam Jakubowski Zbigniew Stanisław Szewczak

We consider a class of strongly mixing sequences with infinite second moment. This class contains important GARCH processes that are applied in econometrics. We show the relative stability for such processes and construct a counterexample. We apply these results and obtain a new CLT without the requirement of exponential decay of mixing coefficients, and provide a counterexample to this as well.

Foundations doi: 10.3390/foundations5040032

Authors: Angelo De Santis Roberto Dini Gianfranco Cianchini

This study investigates the origin and amplification of magnetic fields in planets and galaxies, emphasizing the foundational role of a seed magnetic field (SMF) in enabling dynamo processes. We propose a universal mechanism whereby an SMF arises naturally in systems where an orbiting body rotates non-synchronously with respect to its central mass. Based on this premise, we derive a general equation for the SMF applicable to both planetary and galactic scales. Incorporating parameters such as orbital distance, rotational velocity, and core radius, we then introduce a dimensionless factor to characterize the amplification of this seed field via dynamo processes. By comparing model predictions with magnetic field data from the solar system and the Milky Way, we find that the observed magnetic fields can be interpreted as the product of a universal gravitationally induced SMF and a body-specific amplification factor. Our results offer a novel perspective on the generation of magnetic fields in a wide range of astrophysical contexts and suggest new directions for theoretical investigation, including the environments surrounding black holes.

Foundations doi: 10.3390/foundations5030031

Authors: Alexander Bershadskii

Using the results of numerical simulations and solar observations, this study shows that the transition from deterministic chaos to hard turbulence in the magnetic field generated by the emerging small-scale, near-surface (within the Sun’s outer 5–10% convection zone) solar MHD dynamos occurs through a randomization process. This randomization process has been described using the concept of distributed chaos, and the main parameter of distributed chaos β has been employed to quantify the degree of randomization (the wavenumber spectrum characterising distributed chaos has a stretched exponential form E(k)∝exp−(k/kβ)β). The dissipative (Loitsianskii and Birkhoff–Saffman integrals) and ideal (magnetic helicity) magnetohydrodynamic invariants govern the randomization process and determine the degree of randomization 0<β≤1 at various stages of the emerging MHD dynamos, directly or through Kolmogorov–Iroshnikov phenomenology (the magnetoinertial range of scales as a precursor of hard turbulence). Despite the considerable differences in the scales and physical parameters, the results of numerical simulations are in quantitative agreement with solar observations (magnetograms) within this framework. The Hall magnetohydrodynamic dynamo is also briefly discussed in this context.

Foundations doi: 10.3390/foundations5030030

Authors: Thomas Schürmann

We revisit the Extended Uncertainty Principle (EUP) from an operational viewpoint, replacing wavefunction-based widths with apparatus-defined position constraints such as a finite slit of width Δx or a geodesic ball of radius R. Using Hermitian momentum operators consistent with the EUP algebra, we prove a sharp lower bound on the product of momentum spread and preparation size in one dimension and show that it reduces smoothly to the standard quantum limit as the deformation vanishes. We then extend the construction to dimensions two and three on spaces of constant curvature and obtain the corresponding bound for spherical confinement, clarifying its geometric meaning via an isometry to S2 and S3. The framework links curvature-scale effects to operational momentum floors and suggests concrete tests in diffraction, cold-atom, and optomechanical settings.

Foundations doi: 10.3390/foundations5030029

Authors: Everett M. Criss Anne M. Hofmeister

The often used analytical representation of the Maxwell–Boltzmann classical speed distribution function (F) for elastic, indivisible particles assumes an infinite limit for the speed. Consequently, volume and the number of particles (n) extend to infinity: Both infinities contradict assumptions underlying this non-relativistic formulation. Finite average kinetic energy and temperature (T) result from normalization of F removing n: However, total energy (i.e., heat of the collection) remains infinite because n is infinite. This problem persists in recent adaptations. To better address real (finite) systems, wherein T depends on heat, we generalize this one-parameter distribution (F, cast in energy) by proposing a two-parameter gamma distribution function (F*) in energy which reduces to F at large n. Its expectation value of kT (k = Boltzmann’s constant) replicates F, whereas the shape factor depends on n and affects the averages, as expected for finite systems. We validate F* via a first-principle, molecular dynamics numerical model of energy and momentum conserving collisions for 26, 182, and 728 particles in three-dimensional physical space. Dimensionless calculations provide generally applicable results; a total of 107 collisions suffice to represent an equilibrated collection. Our numerical results show that individual momentum conserving collisions in three-dimensions provide symmetrical speed distributions in all Cartesian directions. Thus, momentum and energy conserving collisions are the physical cause for equipartitioning of energy: Validity of this theorem for other systems depends on their specific motions. Our numerical results set upper limits on kinetic energy of individual particles; restrict the n particles to some finite volume; and lead to a formula in terms of n for conserving total energy when utilizing F* for convenience. Implications of our findings on matter under extreme conditions are briefly discussed.

Foundations doi: 10.3390/foundations5030028

Authors: Maurice P. Schmitt Andrey Milchev

Knots are ubiquitous in polymers and biological macromolecules such as DNA and proteins, yet their behavior and functionality are still not sufficiently explored. Here we investigate the impact of Poiseuille flow on simple knots in flexible polymers placed in a quasi-rectangular micro-channel by systematically varying the flow strength for different chain lengths. Hydrodynamic interactions are accounted for by means of Multi-Particle Collision Dynamics (MPCD). We find that initially loosely localized knots in polymer coils typically tighten under shear to several segments beyond a certain body force threshold. At higher shear rates, intermittent transition from chain stretching to tumbling is observed which correlates with strong fluctuations in the knot size. Somewhat unexpectedly, our results indicate that the influence of channel width on tightening steadily increases with growing width even at equal mean shear rate γ˙¯.

Foundations doi: 10.3390/foundations5030027

Authors: Antonio E. Bargellini Daniele Ritelli Giulia Spaletta

The aim of this work was to construct explicit expressions for the summation of Dirichlet Beta functions with odd arguments and Zeta functions with even arguments. In the established literature, this is typically done using Fourier series expansions or Bernoulli numbers and polynomials. Here, instead, we achieve our goal by employing tools from probability: specifically, we introduce a generalisation of a technique based on multiple integrals and the algebra of random variables. This also allows us to increase the number of nested integrals and Cauchy random variables involved. Another key contribution is that, by generalising the exponent of Cauchy random variables, we obtain an original proof of the reflection formulae for the Digamma and Trigamma functions. These probabilistic proofs crucially utilise the Mellin transform to compute the integrals needed to determine probability density functions. It is noteworthy that, while understanding the presented topic requires knowledge of the rules for calculating multiple integrals (Fubini’s Theorem) and the algebra of continuous random variables, these are concepts commonly acquired by second-year university students in STEM disciplines. Our study thus offers new perspectives on how the mathematical functions considered relate and shows the significant role of probabilistic methods in promoting comprehension of this research area, in a way accessible to a broad and non-specialist audience.

Foundations doi: 10.3390/foundations5030026

Authors: Fabio Sattin Dominique Franck Escande

Differential equations need boundary conditions (BCs) for their solution. It is widely acknowledged that differential equations and BCs are representative of independent physical processes, and no correlations between them are required. Two recent studies by Hilhorst, Chung et al. argue instead that, in the specific case of diffusion equations (DEs) in bounded systems, BCs are uniquely constrained by the form of transport coefficients. In this paper, we revisit how DEs emerge as fluid limits out of a picture of stochastic transport. We point out their limits of validity and argue that, in most physical systems, BCs and DEs are actually uncorrelated by virtue of the failure of diffusive approximation near the system’s boundaries. When, instead, the diffusive approximation holds everywhere, we show that the correct chain of reasoning goes in the direction opposite to that conjectured by Hilhorst and Chung: it is the choice of the BCs that determines the form of the DE in the surroundings of the boundary.

Foundations doi: 10.3390/foundations5030025

Authors: Angelo De Santis Roberto Dini

We introduce the concept of the magnetic metapole—a theoretical extension of classical multipole theory involving a fractional j pole count (related to the harmonic degree n as j = 2n). Defined by a scalar potential with colatitudinal dependence and no radial variation, the metapole yields a magnetic field that decays as 1/r and is oriented along spherical surfaces. Unlike classical multipoles, the metapole cannot be described as a point source; rather, it corresponds to an extended or filamentary magnetic distribution as derived from Maxwell’s equations. We demonstrate that pairs of oppositely oriented metapoles (up/down) can, at large distances, produce magnetic fields resembling those of classical monopoles. A regularized formulation of the potential resolves singularities for the potential and the field. When applied in a bounded region, it yields finite field energy, enabling practical modeling applications. We propose that the metapole can serve as a conceptual and computational framework for representing large-scale magnetic field structures particularly where standard dipole-based models fall short. This construct may have utility in both geophysical and astrophysical contexts, and it provides a new tool for equivalent source modeling and magnetic field decomposition.

Foundations doi: 10.3390/foundations5030024

Authors: Marco Favretti

In this paper, we review Helmholtz–Boltzmann thermodynamics for mechanical systems depending on parameters, and we compute the Fisher information matrix for the associated probability density. The divergence of Fisher information has been used as a signal for the existence of phase transitions in finite systems even in the absence of a thermodynamic limit. We investigate through examples if qualitative changes in the dynamic of mechanical systems described by Helmholtz–Boltzmann thermodynamic formalism can be detected using Fisher information.

Foundations doi: 10.3390/foundations5030023

Authors: Zeqian Chen

This paper presents a mathematical formalism for generative artificial intelligence (GAI). Our starting point is an observation that a “histories” approach to physical systems agrees with the compositional nature of deep neural networks. Mathematically, we define a GAI system as a family of sequential joint probabilities associated with input texts and temporal sequences of tokens (as physical event histories). From a physical perspective on modern chips, we then construct physical models realizing GAI systems as open quantum systems. Finally, as an illustration, we construct physical models realizing large language models based on a transformer architecture as open quantum systems in the Fock space over the Hilbert space of tokens. Our physical models underlie the transformer architecture for large language models.

Foundations doi: 10.3390/foundations5020022

Authors: Marie-Christine Firpo

In axisymmetric fusion devices like tokamaks, the winding of the magnetic field is characterized by its safety profile q=qB. Similarly, the winding of the current density field is characterized by qJ. Currently, the relationship between qB and qJ profiles and their effect on tokamak plasma confinement properties remains unexplored, as the qJ profile is neither computed nor considered. This study presents a reconstruction of the current density winding profile from experimental data in the quiescent H-mode. The topology analysis derived from (qB,qJ) was carried out using Hamada coordinates. It shows a large central plasma region unaffected by current filamentation-driven resonant magnetic perturbations, while the outer region harbors a spectrum of magnetic resonant modes, induced by current filaments located within the core plasma, which degrade peripheral confinement. These results suggest a QH-mode signature pattern needing further validation with additional data. Implementing (qB,qJ) real-time monitoring could provide insights into tokamak confinement regimes with significant implications.

Foundations doi: 10.3390/foundations5020021

Authors: Alexander Bershadskii

Using the results of numerical simulations and astrophysical observations (mainly in the WMAP and Planck frequency bands), it is shown that Galactic foreground emission becomes more sensitive to the mean magnetic field with the frequency, resulting in the appearance of two levels of its randomization due to the chaotic/turbulent dynamics of a magnetized interstellar medium dominated by magnetic helicity. The galactic foreground emission is more randomized at higher frequencies. The Galactic synchrotron and polarized dust emissions have been studied in detail. It is shown that the magnetic field imposes its level of randomization on the synchrotron and dust emission. The main method for the theoretical consideration used in this study is the Kolmogorov–Iroshnikov phenomenology in the frames of distributed chaos notion. Despite the vast differences in the values of physical parameters and spatio-temporal scales between the numerical simulations and the astrophysical observations, there is a quantitative agreement between the results of the astrophysical observations and the numerical simulations in the frames of the distributed chaos notion.

Foundations doi: 10.3390/foundations5020020

Authors: Alexander Robitzsch

Mean–geometric mean (MGM) linking compares group differences on a latent variable θ within the two-parameter logistic (2PL) item response theory model. This article investigates three specifications of MGM linking that differ in the weighting of item difficulty differences: unweighted (UW), discrimination-weighted (DW), and precision-weighted (PW). These methods are evaluated under conditions where random DIF effects are present in either item difficulties or item intercepts. The three estimators are analyzed both analytically and through a simulation study. The PW method outperforms the other two only in the absence of random DIF or in small samples when DIF is present. In larger samples, the UW method performs best when random DIF with homogeneous variances affects item difficulties, while the DW method achieves superior performance when such DIF is present in item intercepts. The analytical results and simulation findings consistently show that the PW method introduces bias in the estimated group mean when random DIF is present. Given that the effectiveness of MGM methods depends on the type of random DIF, the distribution of DIF effects was further examined using PISA 2006 reading data. The model comparisons indicate that random DIF with homogeneous variances in item intercepts provides a better fit than random DIF in item difficulties in the PISA 2006 reading dataset.

Foundations doi: 10.3390/foundations5020019

Authors: Paul M. Alsing Richard J. Birrittella

We show numerical evidence for a bipartite d×d pure state entanglement witness that is readily calculated from the wavefunction coefficients directly, without the need for the numerical computation of eigenvalues. This is accomplished by using an approximate analytic diagonalization of the bipartite state that captures dominant contributions to the negativity of the partially transposed state. We relate this entanglement witness to the Log Negativity, and show that it exactly agrees with it for the class of pure states whose quantum amplitudes form a positive Hermitian matrix. In this case, the Log Negativity is given by the negative logarithm of the purity of the amplitudes considered a density matrix. In other cases, the witness forms a lower bound to the exact, numerically computed Log Negativity. The formula for the approximate Log Negativity achieves equality with the exact Log Negativity for the case of an arbitrary pure state of two qubits, which we show analytically. We compare these results to a witness of entanglement given by the linear entropy. Finally, we explore an attempt to extend these pure state results to mixed states. We show that the Log Negativity for this approximate formula is exact for the class of pure state decompositions, for which the quantum amplitudes of each pure state form a positive Hermitian matrix.

Foundations doi: 10.3390/foundations5020018

Authors: Sotiris K. Ntouyas Bashir Ahmad Jessada Tariboon

In this survey, we have included the recent results on Lyapunov-type inequalities for differential equations of fractional order associated with Dirichlet, nonlocal, multi-point, anti-periodic, and discrete boundary conditions. Our results involve a variety of fractional derivatives such as Riemann–Liouville, Caputo, Hilfer–Hadamard, ψ-Riemann–Liouville, Atangana–Baleanu, tempered, half-linear, and discrete fractional derivatives.

Foundations doi: 10.3390/foundations5020017

Authors: Angelo Morro

This paper revisits some aspects connected with the methods for the determination of thermodynamically consistent models. While the concepts apply to the general context of continuum physics, the details are developed for the modelling of deformable dielectrics. The symmetry condition arising from the balance of angular momentum is viewed as a constraint for the constitutive equations and is shown to be satisfied by sets of objective fields that account jointly for deformation and electric field. The second law of thermodynamics is considered in a generalized form where the entropy production is given by a constitutive function possibly independent of the other constitutive functions. Furthermore, a representation formula is applied for solving the Clausius–Duhem inequality with respect to the chosen unknown fields.

Foundations doi: 10.3390/foundations5020016

Authors: Ruth E. Kastner

I dispute the conventional claim that the second law of thermodynamics is saved from a “Maxwell’s demon” by the entropy cost of information erasure and show that instead it is measurement that incurs the entropy cost. Thus, Brillouin, who identified measurement as savior of the second law, was essentially correct, and putative refutations of his view, such as Bennett’s claim to measure without entropy cost, are seen to fail when the applicable physics is taken into account. I argue that the tradition of attributing the defeat of Maxwell’s demon to erasure rather than to measurement arose from unphysical classical idealizations that do not hold for real gas molecules, as well as a physically ungrounded recasting of physical thermodynamical processes into computational and information-theoretic conceptualizations. I argue that the fundamental principle that saves the second law is the quantum uncertainty principle applying to the need to localize physical states to precise values of observables in order to effect the desired disequilibria aimed at violating the second law. I obtain the specific entropy cost for localizing a molecule in the Szilard engine and show that it coincides with the quantity attributed to Landauer’s principle. I also note that an experiment characterized as upholding an entropy cost of erasure in a “quantum Maxwell’s demon” actually demonstrates an entropy cost of measurement.

Foundations doi: 10.3390/foundations5020015

Authors: Edward Bormashenko

Ramsey analysis is applied to the problem of the relativistic and quantum synchronization of clocks. Various protocols of synchronization are addressed. Einstein and Eddington special relativity synchronization procedures are considered, and quantum synchronization is discussed. Clocks are seen as the vertices of the graph. Clocks may be synchronized or unsynchronized. Thus, introducing complete, bi-colored, Ramsey graphs emerging from the lattices of clocks becomes possible. The transitivity of synchronization plays a key role in the coloring of the Ramsey graph. Einstein synchronization is transitive, while general relativity and quantum synchronization procedures are not. This fact influences the value of the Ramsey number established for the synchronization graph arising from the lattice of clocks. Any lattice built of six clocks, synchronized with quantum entanglement, will inevitably contain the mono-chromatic triangle. The transitive synchronization of logical clocks is discussed. Interrelation between the symmetry of the clock lattice and the structure of the synchronization graph is addressed. Ramsey analysis of synchronization is important for the synchronization of computers in networks, LIGO, and Virgo instruments intended for the registration of gravitational waves and GPS tame-based synchronization.

Foundations doi: 10.3390/foundations5020014

Authors: Matteo Giordano

Parameter identification problems in partial differential equations (PDEs) consist in determining one or more functional coefficient in a PDE. In this article, the Bayesian nonparametric approach to such problems is considered. Focusing on the representative example of inferring the diffusivity function in an elliptic PDE from noisy observations of the PDE solution, the performance of Bayesian procedures based on Gaussian process priors is investigated. Building on recent developments in the literature, we derive novel asymptotic theoretical guarantees that establish posterior consistency and convergence rates for methodologically attractive Gaussian series priors based on the Dirichlet–Laplacian eigenbasis. An implementation of the associated posterior-based inference is provided and illustrated via a numerical simulation study, where excellent agreement with the theory is obtained.

Foundations doi: 10.3390/foundations5020013

Authors: Kamran Azhar Asim Nadeem Yilun Shang

In this paper, we study the fault-tolerant metric dimension in graph theory, an important measure against failures in unique vertex identification. The metric dimension of a graph is the smallest number of vertices required to uniquely identify every other vertex based on their distances from these chosen vertices. Building on existing work, we explore fault tolerance by considering the minimal number of vertices needed to ensure that all other vertices remain uniquely identifiable even if a specified number of these vertices fails. We compute the fault-tolerant metric dimension of various chemical graphs, namely fullerenes, benzene, and polyphenyl graphs.

Foundations doi: 10.3390/foundations5020012

Authors: Reinout Heijungs

Vectors are almost always introduced as objects having magnitude and direction. Following that idea, textbooks and courses introduce the concept of a vector norm and the angle between two vectors. While this is correct and useful for vectors in two- or three-dimensional Euclidean space, these concepts make no sense for more general vectors, that are defined in abstract, non-metric vector spaces. This is even the case when an inner product exists. Here, we analyze how several textbooks are imprecise in presenting the restricted validity of the expressions for the norm and the angle. We also study one concrete example, the so-called ‘vector-based sustainability analytics’, in which scientists have gone astray by mistaking an abstract vector for a Euclidean vector. We recommend that future textbook authors introduce the distinction between vectors that have and that do not have magnitude and direction, even in cases where an inner product exists.

Foundations doi: 10.3390/foundations5020011

Authors: Ioannis G. Tsoulos Vasileios Charilogis Dimitrios Tsalikakis

Artificial neural networks are widely used in applications from various scientific fields and in a multitude of practical applications. In recent years, a multitude of scientific publications have been presented on the effective training of their parameters, but in many cases overfitting problems appear, where the artificial neural network shows poor results when used on data that were not present during training. This text proposes the incorporation of a three-stage evolutionary technique, which has roots in the differential evolution technique, for the effective training of the parameters of artificial neural networks and the avoidance of the problem of overfitting. The new method effectively constructs the parameter value range of the artificial neural network with one processing level and sigmoid outputs, both achieving a reduction in training error and preventing the network from experiencing overfitting phenomena. This new technique was successfully applied to a wide range of problems from the relevant literature and the results were extremely promising. From the conducted experiments, it appears that the proposed method reduced the average classification error by 30%, compared to the genetic algorithm, and the average regression error by 45%, as compared to the genetic algorithm.

Foundations doi: 10.3390/foundations5010010

Authors: David Ellerman

Category theory has foundational importance because it provides conceptual lenses to characterize what is important and universal in mathematics—with adjunction seeming to be the primary lens. Our topic is a theory showing “where adjoints come from”. The theory is based on object-to-object “chimera morphisms”, “heteromorphisms”, or “hets” between the objects of different categories (e.g., the insertion of generators as a set-to-group map). After showing that heteromorphisms can be treated rigorously using the machinery of category theory (bifunctors), we show that all adjunctions between two categories arise (up to an isomorphism) as the representations (i.e., universal models) within each category of the heteromorphisms between the two categories. The conventional treatment of adjunctions eschews the whole concept of a heteromorphism, so our purpose is to shine a new light on this notion by showing its origin as a het between categories being universally represented within each of the two categories. This heteromorphic treatment of adjunctions shows how they can be split into two separable universal constructions. Such universals can also occur without being part of an adjunction. We conclude with the idea that it is the universal constructions (adjunctions being an important special case) that are really the foundational concepts to pick out what is important in mathematics and perhaps in other sciences, not to mention in philosophy.

Foundations doi: 10.3390/foundations5010009

Authors: Koun Shirai

In glass physics, order parameters have long been used in the thermodynamic description of glasses, but their nature is not yet clear. The difficulty is how to find order in disordered systems. This study provides a coherent understanding of the nature of order parameters for glasses and crystals, starting from the fundament of the definition of state variables in thermodynamics. The state variable is defined as the time-averaged value of a dynamical variable under the constraints, when equilibrium is established. It gives the same value at any time it is measured as long as the equilibrium is maintained. From this definition, it is deduced that the state variables of a solid are the time-averaged positions of all atoms constituting the solid, and the order parameters are essentially the same as state variables. Therefore, the order parameters of a glass are equilibrium atom positions.

Foundations doi: 10.3390/foundations5010008

Authors: Miloslav Znojil

In spite of its unbroken PT symmetry, the popular imaginary cubic oscillator Hamiltonian H(IC)=p2+ix3 does not satisfy all of the necessary postulates of quantum mechanics. This failure is due to the “intrinsic exceptional point” (IEP) features of H(IC) and, in particular, to the phenomenon of a high-energy asymptotic parallelization of its bound-state-mimicking eigenvectors. In this paper, it is argued that the operator H(IC) (and the like) can only be interpreted as a manifestly unphysical, singular IEP limit of a hypothetical one-parametric family of certain standard quantum Hamiltonians. For explanation, ample use is made of perturbation theory and of multiple analogies between IEPs and conventional Kato’s exceptional points.

Foundations doi: 10.3390/foundations5010007

Authors: Mihail Mihaylov Konstantinov Petko Hristov Petkov Ekaterina Borisova Madamlieva

In this paper, we consider the application of brute force computational techniques (BFCTs) for solving computational problems in mathematical analysis and matrix algebra in a floating-point computing environment. These techniques include, among others, simple matrix computations and the analysis of graphs of functions. Since BFCTs are based on matrix calculations, the program system MATLAB® is suitable for their computer realization. The computations in this paper are completed in double precision floating-point arithmetic, obeying the 2019 IEEE Standard for binary floating-point calculations. One of the aims of this paper is to analyze cases where popular algorithms and software fail to produce correct answers, failing to alert the user. In real-time control applications, this may have catastrophic consequences with heavy material damage and human casualties. It is known, or suspected, that a number of man-made catastrophes such as the Dharhan accident (1991), Ariane 5 launch failure (1996), Boeing 737 Max tragedies (2018, 2019) and others are due to errors in the computer software and hardware. Another application of BFCTs is finding good initial guesses for known computational algorithms. Sometimes, simple and relatively fast BFCTs are useful tools in solving computational problems correctly and in real time. Among particular problems considered are the genuine addition of machine numbers, numerically stable computations, finding minimums of arrays, the minimization of functions, solving finite equations, integration and differentiation, computing condensed and canonical forms of matrices and clarifying the concepts of the least squares method in the light of the conflict remainders vs. errors. Usually, BFCTs are applied under the user’s supervision, which is not possible in the automatic implementation of computational methods. To implement BFCTs automatically is a challenging problem in the area of artificial intelligence and of mathematical artificial intelligence in particular. BFCTs allow to reveal the underlying arithmetic in the performance of computational algorithms. Last but not least, this paper has tutorial value, as computational algorithms and mathematical software are often taught without considering the properties of computational algorithms and machine arithmetic.

Foundations doi: 10.3390/foundations5010006

Authors: Felipe J. Carmona-Moreno Armando Gallegos José J. Barba-Franco Ernesto Urenda-Cázares Enrique V. Jiménez-Guerrero Jorge E. Macías-Díaz

Low-grade gliomas are a group of brain tumors that mostly affect people in early adulthood. A glioma is a tumor that originates in glial cells. They are classified into four levels according to their level of proliferation, with grades 1 and 2 called low-grade gliomas. In this research, we conduct an analysis focused on a mathematical model that emulates the behavior of low-grade gliomas with chemotherapy treatment based on a system of nonlinear differential equations. An analysis of the model is carried out in the absence of treatment, resulting in a predictive solution for the behavior of glioma if it left untreated for some reason. In turn, a stability analysis is carried out on the system of equations to find the critical points for treating glioma. In addition, the numerical results of the model are obtained, presenting the state variables at each instant. Finally, some simulations are presented, varying the moments of treatment initiation and the applied doses of Temozolomide.

Foundations doi: 10.3390/foundations5010005

Authors: Helena Cristina Vasconcelos Maria Meirelles Reşit Özmenteş Luís Santos

This study employs advanced structural characterization techniques, including anomalous small-angle X-ray scattering (ASAXS), small-angle X-ray scattering (SAXS), and X-ray photoelectron spectroscopy (XPS), to investigate erbium (Er3+)-doped silica-based glass-ceramic thin films synthesized via the sol–gel method. This research examines the SiO2-TiO2 and SiO2-TiO2-PO2.5 systems, focusing on the formation, dispersion, and structural integration of Er3+-containing nanocrystals within the amorphous matrix under different thermal treatments. Synchrotron radiation tuned to the LIII absorption edge of erbium enabled ASAXS measurements, providing element-specific details about the localization of Er3+ ions. The findings confirm their migration into crystalline phases, such as erbium phosphate (EPO) and erbium titanate (ETO). SAXS and Guinier analysis quantified nanocrystal sizes, revealing trends influenced by their composition and heat treatment. Complementary XPS analysis of the Er 5p core-level states provided detailed information on the chemical and electronic environment of the Er3+ ions, confirming their stabilization within the crystalline structure. Transmission electron microscopy (TEM) highlighted the nanoscale morphology, verifying the aggregation of Er3+ ions into well-defined nanocrystals. The results offer a deeper understanding of their size, distribution, and interaction with the surrounding matrix.

Foundations doi: 10.3390/foundations5010004

Authors: Alexander Robitzsch

In fixed item parameter calibration (FIPC), an item response theory (IRT) model is estimated with item parameters fixed at reference values to estimate the distribution parameters within a specific group. The presence of random differential item functioning (DIF) within this group introduces additional variability in the distribution parameter estimates, which is captured by the linking error (LE). Conventional LE estimates, based on item jackknife methods, are subject to positive bias due to sampling errors. To address this, this article introduces a bias-corrected LE estimate. Moreover, the use of statistical inference is examined using the newly proposed bias-corrected total error, which includes both the sampling error and LE. The proposed error estimates were evaluated through a simulation study, and their application is illustrated using PISA 2006 data for the reading domain.

Foundations doi: 10.3390/foundations5010003

Authors: Angelo Plastino Diana Monteoliva

In this study, we explore the order–disorder transition in the dynamics of a straightforward master equation that describes the evolution of a probability distribution between two states, p1 and p2 (with p1+p2=1). We focus on (1) the behavior of entropy S, (2) the distance D from the uniform distribution (p1=p2=1/2), and (3) the free energy F. To facilitate understanding, we introduce two price-ratios: ηS=dS/dtdF/dt and ηD=dD/dtdF/dt. They respectively define the energetic costs of modifying (1) S and (2) D. Our findings indicate that both energy costs diverge to plus and minus infinity as the system approaches the uniform distribution, marking a critical transition point where the master equation temporarily loses its physical meaning. Following this divergence, the system stabilizes itself into a new well-behaved regime, reaching finite values that signify a new steady state. This two-regime behavior showcases the intricate dynamics of simple probabilistic systems and offers valuable insights into the relationships between entropy, distance in probability space, and free energy within the framework of statistical mechanics, making it a useful case study that highlights the underlying principles of the system’s evolution and equilibrium. Our discussion revolves about the order–disorder contrast that is important in various scientific disciplines, including physics, chemistry, and material science, and even in broader contexts like philosophy and social sciences.

Foundations doi: 10.3390/foundations5010002

Authors: Alexander Robitzsch

Stocking–Lord (SL) linking is a widely used linking method based on item response theory (IRT). This article examines the variability in SL linking parameter estimates within the two-parameter logistic (2PL) model. The uncertainty in SL linking arises from the sampling variability (standard error) and item selection (linking error), which can induce variability due to random differential item functioning (DIF). Three linking error estimation approaches are compared in this paper: the conventional jackknife linking error method, a newly developed approximate jackknife linking error method, and a Taylor approximation-based estimate. Simulation studies showed that the approximate jackknife method closely aligns with the traditional jackknife linking error method and outperforms the linking error estimation approach based on Taylor approximation. The adequacy of coverage rates for SL linking parameter estimates was also assessed using estimates of the total error. Results from a simulation study demonstrate that the bias-corrected total error provides superior coverage rates compared to both the conventional total error and the standard error, which does not account for item-related uncertainty due to random DIF.

Foundations doi: 10.3390/foundations5010001

Authors: Pierre Jamet Aurélien Drezet

In this work, we present a new theoretical approach to interpreting and reproducing quantum mechanics using trajectory-guided wavelets. Inspired by the 1925 work of Louis de Broglie, we demonstrate that pulses composed of a difference between a delayed wave and an advanced wave (known as antisymmetric waves) are capable of following quantum trajectories predicted by the de Broglie–Bohm theory (also known as Bohmian mechanics). Our theory reproduces the main results of orthodox quantum mechanics and unlike Bohmian theory, is local in the Bell sense. We show that this is linked to the superdeterminism and past–future (anti)symmetry of our theory.

Foundations doi: 10.3390/foundations4040046

Authors: Jeffery A. Secrest Daniel Jones

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.

Foundations doi: 10.3390/foundations4040045

Authors: Moses Kayanda Kiteto Cleophas Achisa Mecha

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.

Foundations doi: 10.3390/foundations4040044

Authors: Salvador Cruz Rambaud

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−a1)n1(x−a2)n2⋯(x−ak)nk, where a1,a2,⋯,ak∈R and n1,n2,⋯,nk∈N. A first approach to this issue is the Taylor expansion of the dividend P(x) at a point a, which clearly shows the quotient and the remainder of the division of P(x) by (x−a)k, where the degree of P(x), 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.

Foundations doi: 10.3390/foundations4040043

Authors: Charles-Gérard Lucas Jérôme Gilles

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.

Foundations doi: 10.3390/foundations4040042

Authors: Jinjie Liu Qi Lu Hacene Boukari Fatima Boukari

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 c2. When c2 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 c2 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.

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’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.

Foundations doi: 10.3390/foundations4040040

Authors: Carlos Sosa Henríquez Martynas Lendraitis

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.

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–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 N(ux), where ux 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 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–De Vries equation, the Kuramoto–Sivashinsky equation, the generalized Korteweg–De Vries–Kuramoto–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≥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.

Foundations doi: 10.3390/foundations4040036

Authors: Dimitris M. Christodoulou Demosthenes Kazanas

We consider spontaneous quark transitions between the Λ0 baryon and its resonant states, and (anti)quark transitions between the neutral kaon K0 and the two heavy ηq-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 Λc+-baryon. We find that the flips s→c and s¯→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 d¯-ground. The wider quark range stems from the flip s→b, which costs 283 MeV more (or 3× more) than the corresponding antiquark flip s¯→b¯. At the same time, transitions from the respective ground states to the s and s¯ states (or the c and 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 Λ-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’ 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 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’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–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.

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’s multiple t-values, Kaneko and Tsumura’s multiple T-values, and Xu and this paper’s author’s multiple S-values. Xu and this paper’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–Goldstein-type relations for the generalized Mehler–Fock transform initially introduced by B. L. J. Braaksma and B. M. Meulenbeld (Compositio Math., 18(3):235–287, 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–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 ℏc∼Λ/cp for a constant Λ>0, and p>1. We show how the limit c→∞ implies classicality in quantum mechanics and explain why p has to be larger than 1. As the limit c→∞ 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–Garnett theory. The other method uses the transformation medium theory and involves the transformation of the material’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’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–ligand docking analyses, highlighting the complexities of inhibitor–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,∞).

Foundations doi: 10.3390/foundations4030022

Authors: Andreas Schlatter

We explicitly calculate the value of the cosmological constant, Λ, 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 Λ~RU−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 “thermal bath” 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’ 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–surfactant–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—sodium dodecyl sulphate (SDS) in alkaline–polymer media—which is composed of a commercial water-soluble polymer (Flopaam AN125SH®, SNF Floerger, Andrézieux-Bouthéon, 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–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 ≠ 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∣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⊆S\Δ(F) for some S∈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∈Sat(F), then we define the Sat(F)-rank of S as the minimum of {cardinality(X)∣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 ‘pore-sifting’ 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–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’s Law), scientific knowledge (Lotka’s Law), people (Auerbach’s Law), and written or verbal information (Zipf’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’s principle. We discuss Mach’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’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&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>40, we observe the appearance of “potential” 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—here from daily, n=1, to “monthly”, 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’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 “space”) and continuous aspects (including “time”) presents another challenge. This study proposes a novel approach named the “Δ table” (delta table), which transforms space–time multivariate data consisting of discrete and continuous variables into a tabular format. The Δ 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’s developed, newly industrialized, and developing economies in recent decades as an example of an attempt to apply the Δ 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’s method (CHN) to solve Banach space valued equations. Our analysis approach is two fold: The first way uses Taylor’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’s methods such as Newton’s, Kou’s and Jarratt’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.