Introductory and intermediate electrical-engineering education commonly models fundamental circuit and device parameters, including inductance, capacitance, resistance, permeability, permittivity, conductivity, characteristic impedance, transformer turns ratio, machine constants, amplifier gain, resonant frequency, propagation velocity, and mutual inductance, as numerical constants. While this approximation is valid within the intended small-signal operating regime, it becomes methodologically incomplete when these coefficients exhibit measurable state dependence. This paper presents a comparative audit of thirteen such coefficients by systematically contrasting their textbook formulations with the established engineering literature and interpreting the results through two complementary frameworks: the JCGM GUM-6:2020 measurement-model methodology for omitted effects, and the Heckmann–Nye/Gasik multidomain coupling architecture for multi-axis physical interactions. The analysis demonstrates that mainstream engineering practice routinely exploits state-dependent coefficients without invoking new physical laws, and that relaxing the constant-coefficient assumption naturally introduces physically meaningful terms, including the inductive contribution IdL/dt, the capacitive counterpart VdC/dt, and the mutual-inductance term i2dM/dt. The principal scientific contribution is the development and experimental validation of a unified theoretical and engineering framework for high-power resonant transformers and orthogonal Metglas AMCC-1000 paraformers. The proposed approach combines a new physics-based modal theory of octave (2:1) parametric excitation with simultaneous optimization of magnetic-core resonance, electrical resonance, nonlinear inductance modulation, resonant conductor lengths selected as integer multiples of the operating resonant wavelength, multi-stranded high-frequency Litz-wire windings, resonant capacitor synthesis, and the nonlinear magnetic characteristics of the AMCC-1000 amorphous core. The modal analysis demonstrates how coupled resonant eigenmodes and engineered state-dependent inductance can be used to satisfy the conditions for stable octave parametric excitation. Experimental results obtained from both the symmetric two-leg resonant transformer and the orthogonal paraformer are in close agreement with analytical predictions and numerical simulations, thereby validating both the proposed electromagnetic design methodology and the underlying modal theory.
The mathematical representation of discrete signals is classically governed by linear basis transforms, such as the Discrete Fourier Transform (DFT), which treat spectral projection as a rigid, deterministic geometric operation. Under this paradigm, signal compression and thresholding rely on heuristic error metrics like the Minimum Mean Squared Error (MMSE). To mathematically axiomatize these fundamental operations, this paper reformulates computational basis transforms through the lens of Cooperative Game Theory. By defining discrete signal reconstruction as a Grand Coalition of orthogonal frequency players, we establish a strict mathematical isomorphism between functional analysis and cooperative game theory. We prove that the spectral energy assigned to each frequency coefficient is exactly its Shapley Value and that classical MMSE minimization is mathematically equivalent to maximizing the retained Shapley payout. Furthermore, we extend this framework to physical hardware and linear shift-invariant (LSI) systems, modeling 8-bit quantization and the Modulation Transfer Function (MTF) as sub-additive “economic taxes” on the coalition. Finally, by intentionally violating the Shapley Symmetry Axiom to mimic the Contrast Sensitivity Function (CSF) of the human visual system, we propose a fundamentally new mathematical basis: the Perceptual Game-Theoretic Transform (PGTT). Unlike classical methods that rely on post hoc quantization for signal compression, the PGTT acts as an inherently efficient transform that structurally guarantees sub-Nyquist computational complexity and dynamic range reallocation prior to physical hardware saturation.
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.
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.
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.
A nonlinear backstepping control framework is developed for autonomous landing of a quadrotor on a wave-excited marine platform. This study addresses the underactuated nature of the aerial vehicle and the strong coupling between translational and rotational dynamics, ensuring stable trajectory tracking under sea-induced disturbances. Reference trajectories are generated through physically grounded Pierson–Moskowitz (PM) and modified Pierson–Moskowitz (MPM) wave spectra, enabling realistic modeling of vertical heave motion, while horizontal position and yaw are defined through harmonic components adapted to the sea-state regime. The controller is designed through a seven-step recursive backstepping procedure, with Lyapunov functions guaranteeing asymptotic stability of the tracking errors for the regulated outputs. A modular MATLAB simulation platform is implemented, integrating the full six-DOF quadrotor dynamics, the control algorithm, and spectral reference generation. Numerical simulations demonstrate that the Lyapunov function derivatives remain negative over the entire simulation horizon, confirming asymptotic convergence. Comparative results with a tuned PID (proportional integral derivative) controller indicate superior tracking performance and damping and reduced amplitude and phase errors for the backstepping approach, especially under MPM-based trajectories representing rough sea states. The proposed framework establishes a reliable basis for adaptive extensions and future hardware-in-the-loop validation of autonomous landing on moving marine platforms.
In recent years, large-scale optimization problems have become increasingly common in various fields, such as machine learning and data analysis, generating increased references to both computational cost and accurate solutions. The Grey Wolf Optimizer (GWO) is an efficient collective intelligence algorithm; however, its performance may be limited when high-permissive problems are allowed or in environments with strong multimodal landscapes. In this paper, we propose the Parallel Balanced Grey Wolf Optimizer (ParallelBGWO), a parallel extension of GWO that aims to improve the balance between exploration and exploitation of the search space. The proposed algorithm divides the total population into multiple subpopulations, which cooperate through information exchange. This cooperation reduces the probability of premature convergence and enhances the global search. The parallel implementation exploits modern multi-core computing architectures, achieving a significant reduction in execution time, while at the same time maintaining or improving the quality of the final solutions. Experimental evaluations on high-dimensional benchmark functions show that ParallelBGWO exhibits faster convergence and a reduced number of objective function evaluations compared to the classical version of GWO and other prominent methods. The results highlight ParallelBGWO as an efficient approach for demanding global optimization problems.
We study the extremal behavior of real two-term linear combinations of third-order Zernike modes on the closed unit disk D2. Our primary focus is the classification of interior local extrema; absolute extrema on the boundary circle are treated only where they admit a concise description or as a separate boundary-optimization problem. These modes arise naturally in Zernike expansions of optical wavefront aberrations. For each of the six unordered pairwise linear combinations of third-order modes, we classify the interior local extrema in terms of the two real coefficients. The trefoil–trefoil case is treated more generally through linear combinations of primary n-multifoils; harmonicity and the maximum principle show that no interior local extrema occur and that absolute extrema are attained on the boundary circle. For the remaining pairwise combinations, we give analytic conditions for the existence, uniqueness, location, and values of local extrema, including degenerate exceptional cases. We also compare these local extrema with boundary values and describe the associated absolute-extremum problem on the boundary circle. Symbolic computations are included in the appendix to document several algebraic reductions, and numerical illustrations are provided to visualize the resulting classifications.
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.
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.
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.
This study introduces a novel matrix defined over the nonzero natural numbers, whose entries are governed by a rigorous closed-form expression. The matrix architecture replicates the topological properties of the Ulam spiral, mapping the integer sequence onto a structured lattice with a well-defined formulation. We investigate the interplay between the matrix’s linear algebraic properties and its number-theoretic implications. A primary focus is the established connection between the matrix’s lines, rows, diagonals, and antidiagonals, and the Hardy–Littlewood F-conjecture. By analyzing the matrix’s internal structure, this work provides a new analytical framework for further study of the conjecture. The matrix links its visual characteristics to quadratic polynomials, offering fresh insights into the distribution of prime numbers.
The Clopper–Pearson method of constructing a confidence interval for the probability of success in a binary population that follows a Bernoulli distribution is well known. This paper pedagogically justifies the Clopper–Pearson method and extends the method to all distributions with one parameter whose cumulative probability distribution functions are monotonically continuous with respect to their single parameter. The conservativeness of Clopper–Pearson confidence intervals is proved analytically. Clopper–Pearson confidence intervals are constructed for Poisson, geometric, non-central hyper-geometric, and exponential, etc. It turns out that such extensions result in various well-known confidence intervals in the literature.
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.
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 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 spaces fundamental groups of 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.
We analyze historic S P500 multi-day returns: from daily returns to those accumulated over up to ten days. Despite symmetry breaking between gains and losses in the distribution of returns, resulting in its positive mean and negative skew, realized variance (volatility squared) exhibits remarkably good linear dependence on the number of days of accumulation. Mean of the distribution also shows near perfect linear dependence as well. We analyze this phenomenon both analytically and numerically using a modified Jones-Faddy skew t-distribution.
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.
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.
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.
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.
