Foundations, Volume 6, Issue 1 (March 2026) – 12 articles

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. Full article

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. Full article

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. Full article

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. Full article

Einstein derived the expansion of space ever since the Big Bang started and introduced the possible cosmological constant $\Lambda$. The expansion of space and the present-day expansion rate $H_0$, the Hubble constant, has been discovered by Hubble. Perlmutter discovered the positive value of $\Lambda$, and Zeldovich showed that $\Lambda$ corresponds to the energy density $u_{DE}$ of space. Lamb and Retherford as well as Casimir provided evidence for the idea that $u_{DE}$ might be based on quanta, and Riess et al. provided evidence that $H_0$ 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. Full article

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. Full article

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. Full article

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. Full article

As a contribution to automated set-theoretic inferencing, a translation is proposed of conjunctions of literals of the forms $x=y\setminus z$, $x\ne y\setminus z$, and $z=\left\{x\right\}$, 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 $x=y\setminus z$ and $x\ne y\setminus z$ 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 $\mathsf{NP}$-complete satisfiability problem. Full article

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. Full article

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. Full article