Search Results (126)

The generalized Burgers–Huxley (gBH) equation arises as a canonical model in nerve-pulse propagation (generalizing the Hodgkin–Huxley/FitzHugh–Nagumo excitable-media framework), in population dynamics with Allee-threshold reaction kinetics, and in nonlinear wave propagation in dispersive media; accurate benchmark solutions are essential for quantitative predictions in these domains. We couple the fourth-order exponential time differencing scheme ETDRK4 with a Chebyshev collocation spatial discretization and a linear boundary-lifting procedure to solve the gBH equation on a bounded interval with non-homogeneous Dirichlet data. On the canonical Ismail–Raslan–Rabboh travelling-wave benchmark the scheme attains $L^{\infty }$ errors at the level of floating-point round-off (∼10⁻¹⁹ absolute, ∼10⁻¹⁵ relative) with as few as $N=2$ collocation points and a single time step of size $\Delta t=1.0$—that is, three total nodes and one ETDRK4 advance. In strongly nonlinear regimes ($\gamma =0.1, 0.3, 0.5, 0.9$) the scheme exhibits approximately $\mathcal{O}(\Delta t^{2.45})$ temporal convergence across all four parameter values, consistent with the classical Hochbruck–Ostermann order reduction for exponential integrators on parabolic PDEs with non-homogeneous Dirichlet data. Used as a high-accuracy probe, the scheme provides a diagnostic of independent interest: the wave-speed formula of Wang, Zhu and Lu, still appearing as the exact-solution benchmark in numerical studies as recently as 2020, does not satisfy the partial differential equation. The corrected formula stated by Deng and verified symbolically by Appadu and Tijani is the unique value that makes the travelling-wave ansatz a genuine solution. We derive the residual associated with Wang’s formula in closed form, $R=\frac{\gamma A_1}{2}(A_2-A_2^{\mathrm{W}})(1-v^2)$, and show both analytically and numerically that reported errors for schemes benchmarked against Wang’s formula coincide with the analytical wave-profile gap $\frac{\gamma A_1}{2}|A_2-A_2^{\mathrm{W}}|$ rather than with true scheme accuracy. At the Ismail benchmark this gap equals $3.748\times {10}^{-7}$, which matches the N- and $\Delta t$-independent plateau observed when the scheme is measured against Wang’s profile. In the nerve-pulse and excitable-media interpretation, the two formulas correspond to action-potential propagation speeds of opposite sign at the Ismail benchmark, underscoring that the correction is not a mere algebraic curiosity but changes the qualitative physical prediction of the model. Full article

We present a spectral-action framework for connecting generalized trace dynamics (GTD) to the structural form of the low-energy action of the observed Universe. The fundamental single-STM-atom Lagrangian is decomposed exactly into a purely bosonic sector, boson–fermion cross terms, and bifermionic terms. This sectorwise decomposition furnishes a dictionary to almost-commutative spectral geometry: the bosonic sector supplies a quadratic GTD Dirac functional built from the six split-biquaternionic differential directions together with octonionic vector/gauge fluctuations; the cross-sector supplies, under an explicit localization hypothesis, a sesquilinear fermionic pairing; and the bifermionic sector supplies the scalar/internal channel that is bosonized into the Higgs bridge field. We also record the principal-symbol link between the $\mathrm{SO}(3,3)$ BF variables and the four-dimensional leafwise Dirac operator. The two four-dimensional leaves of the six-dimensional base overlap in two common directions; from the observed (gravitational) leaf, the two nonintersecting directions of the complementary leaf are internal, so the second leaf is reinterpreted as the weak-interaction sector rather than as an independent spacetime—a reinterpretation stated here as an explicit hypothesis. Under stated assumptions—spontaneous localization, Euclidean continuation, six- to four-dimensional BF reduction, and a candidate observed-leaf finite geometry compatible with the $E_6/J_3\left({\mathbb{O}}_{\mathbb{C}}\right)$ inputs—the bosonic heat-kernel expansion yields the structural low-energy classes of terms: Einstein–Hilbert gravity, Yang–Mills kinetic terms, and scalar kinetic and potential terms. In addition, we provide a candidate finite spectral triple with explicit finite trace invariants, verify that the localization map respects the one-generation lepton/quark representation split, identify visible color-singlet scalar channels with electroweak quantum numbers $(\mathbf{1},\mathbf{2},\pm 1/2)$, and exhibit a smooth regulator family with explicit cutoff moments $(f_0,f_2,f_4)$. Conversely, the assembled low-energy spectral action admits a natural inverse bilinear lift back to split bioctonionic trace dynamics. Every arrow of the construction is classified as an exact algebraic identity, an imported result, a working hypothesis, or an open problem. Under this classification, the paper offers a possible architecture for obtaining low-energy gauge–gravity physics from GTD, with conditional consistency checks and reductions; it is not a completed first-principles derivation of the Standard Model coupled to gravity. Full article

Euclidean plane geometry problem solving is a challenging benchmark for artificial intelligence because it requires complex diagram understanding, symbolic deduction, and multi-step reasoning. Constructing effective datasets for this task requires geometric instances that are realizable, non-degenerate, structurally diverse, and paired with human-like proofs. However, existing random or template-based generation pipelines often produce redundant, singular, or infeasible candidates, causing substantial computation to be spent before useful reasoning trajectories can be extracted. To address these limitations, we present FGeo-GCG, a hybrid geometric data synthesis framework built on the FormalGeo-V2 deductive engine. It formulates Geometric Configuration Generation as an incremental linear construction process that decomposes global constraint satisfaction into local construction steps, thereby pruning invalid branches during the generation process. To improve reliability and efficiency, FGeo-GCG combines two validation stages: a safe stochastic Jacobian-rank filter estimates whether local candidate constraints contribute independent algebraic restrictions, and progressive geometric validation checks whether the resulting partial construction remains realizable and non-degenerate. By encoding incidence-, metric-, and symmetry-related dependencies within unified constraint graphs, the framework also connects geometric data synthesis with structural symmetry analysis. Validated constraint graphs are then converted into problem instances through forward deduction, goal decomposition, and multi-dimensional complexity filtering, producing proof targets without manual annotation. Experiments show that the full validation pipeline reduces the failure rate for highly constrained instances. The resulting FGeo-GCG dataset contains more than 50,000 formally validated plane geometric configurations and provides engine-derived reasoning traces and targets for future training and evaluation of neuro-symbolic geometry problem-solving systems. Full article

Binary Waring decomposition seeks to express a homogeneous binary form as a minimal sum of powers of linear forms. In the binary setting, Sylvester’s theorem gives a classical algebraic route for rank determination and parameter recovery through structured Hankel/catalecticant matrices. Although this procedure is exact and interpretable in ideal arithmetic, practical rank identification may become unstable when the input coefficients are contaminated by noise or when the underlying roots are close to degenerate configurations. This paper develops a data-driven rank inference framework coupled with certified Sylvester reconstruction for robust binary Waring decomposition. The proposed method first converts the coefficient sequence into a Hankel-aware graph that captures recurrence-induced dependencies among polynomial coefficients. A graph neural network is then used to infer plausible rank candidates from this structured representation. Instead of accepting a single prediction directly, the framework performs explicit Sylvester reconstruction and algebraic residual verification for candidate ranks. To further improve decision reliability, a lightweight meta-verification module integrates reconstruction residuals, model confidence scores, and stability-related indicators to select the most credible rank. Experiments on large-scale synthetic binary forms show that the proposed meta-guided variant improves rank identification and verified reconstruction success relative to the one-shot hybrid solver under low-to-moderate noise while maintaining the transparency and auditability of classical symbolic–numeric computation. Additional stress tests indicate that performance can degrade under shifted sampling regimes; so, the method should be interpreted as a robust decision layer within the modeled problem class rather than as unconstrained real-world validation. Full article

Accurate design and performance assessment of solar thermal domestic hot water systems coupled with a heat pump auxiliary typically requires transient simulation, as the system’s behavior depends on multiple interactions among collector characteristics, storage stratification, control logic, weather, and draw-off timing. Monthly methods such as the f-chart are useful for first-pass estimates, but they do not resolve stratification, thermostat operation, or demand timing, and they may become inaccurate for stratified thermostat-controlled systems. Direct comparisons of locally inspectable symbolic and black-box surrogate families for this system class remain limited. A 10,982-case development dataset was generated from minute-resolved annual MATLAB simulations, parameterized by collector area, optical efficiency, and first- and second-order loss coefficients. Three surrogate families were benchmarked under a unified protocol, random forest-assisted shape-constrained symbolic regression (SR), feed-forward artificial neural network (ANN) models, and Automatic Learning of Algebraic Models for Optimization (ALAMO), with the f-chart used as a monthly reference method. The targets were the 12 monthly solar fractions under the direct solar heat definition and the corresponding annual mean solar fraction, evaluated on the same independent 991-case test set. SR achieved the lowest average error (mean absolute percentage error, MAPE = 0.82%; root mean square error, RMSE = 0.006), followed by the ANN (MAPE = 2.07%, RMSE = 0.028) and ALAMO (MAPE = 3.67%, RMSE = 0.060), with Nash–Sutcliffe efficiency (NSE) values above 0.98 for all models. Evaluation times were 0.0026–0.124 s per target, compared with about 1000 s for one full-year simulation. These results define the study as a common protocol benchmark within the studied simulator-defined envelope. SR gives the strongest accuracy with local symbolic inspectability, the ANN remains the flexible retrainable option, and ALAMO provides compact algebraic evaluation with the shortest learned model runtime. Full article

We consider the translational–rotational motion of a small axisymmetric body in the gravitational field of two stars of variable masses moving under the influence of their mutual gravitational attraction. The two stars are assumed to lose their masses isotropically with different rates and their total mass decreases according to the joint Meshcherskii law. The relative motion of the stars is described by the corresponding exact solution to Gyldén’s equation and is considered to be given. The small axisymmetric body may change its mass, size and shape while its initial dynamic structure is retained. The problem is analyzed in the framework of Lagrange’s formalism, and equations of translational–rotational motion are derived in terms of the osculating elements of aperiodic motion on quasi-conic sections and the Andoyer variables. As equations of motion of the small body are not integrable, the perturbation theory is applied with the perturbing function expanded into power series in terms of eccentricity and inclination, which are assumed to be small. Averaging these equations over the mean longitudes of the two bodies and two Andoyer angles in the absence of commensurability of frequencies, we obtain the differential equations describing the long-term evolution of the orbital elements and Andoyer variables which may be investigated numerically for different laws of the system parameters’ variation. All the relevant symbolic calculations are performed with the computer algebra system Wolfram Mathematica. Full article

Mobi spaces were introduced by the authors as a possible algebraic axiomatization of spaces in which any two points are connected by a geodesic path. Previous work has focused primarily on the algebraic properties of these structures; here, we return to the original geometric motivation. We present new characterizations of mobi spaces inspired by the important class of smooth manifolds that arise as open subsets of Euclidean (n)-space endowed with a Riemannian metric. We show that such manifolds satisfy the axioms of a mobi space, thereby providing a broad family of natural geometric examples. Full article

This paper studies the algebraic properties of little Hankel operators on Hardy–Sobolev spaces $H_s^2$, focusing on a notion of commutativity defined via adjoint products. For symbols $\phi$ and $\psi$, whose co-analytic parts are trigonometric polynomials, we consider the condition ${\left(H_{\phi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)}={\left(H_{\psi }^{\left(s\right)}\right)}^{*}{H}_{\phi }^{\left(s\right)}\mathrm{on}{H}_s^2.$ It is shown that this adjoint-product commutativity holds if and only if the co-analytic parts of the symbols are real scalar multiples of one another. As a consequence, the commutant of a nonzero Hankel operator on $H_s^2$, within the class of Hankel operators whose co-analytic symbols are trigonometric polynomials, is one-dimensional over $\mathbb{R}$. The proof relies on a direct coefficient analysis exploiting the finite Hankel structure induced by polynomial symbols. The result applies uniformly to all Sobolev exponents $s\ge 0$, including the classical Hardy space $s=0$ and the Dirichlet case $s=1/2$. Full article

We study a generalized class of Jacobsthal–Lucas polynomials that depends on two parameters. First, we introduce essential formulas for these polynomials, involving their series representation, inverse formula, and moment formula. These formulas allow us to investigate this generalized class of polynomials further and to develop novel formulations. The essential standard linearization problem of these polynomials is solved, and the linearization coefficients are given in simple forms. In addition, some mixed linearization formulas with other classes of polynomials are presented. The derivative formulas of these polynomials, expressed as combinations of different polynomials, are given. By employing symbolic algebra methods—most notably Zeilberger’s algorithm and other well-known identities from the literature—many hypergeometric functions appearing in the coefficients can be reduced, resulting in simpler expressions. In addition, some definite integrals are evaluated using the newly introduced formulas. Full article

Differential operators often admit multiple algebraically equivalent symbolic formulations, yet those formulations can differ in the organization of their internal structure prior to solution analysis. A reproducible symbolic framework is introduced to compare such formulations at the level of operator expressions. Within a declared symbolic specification consisting of a fixed grammar, an admissible weight class, canonical compression rules, and an admissible family of reformulations, we define four encoding-relative structural descriptors: structural strain $\tau$, structural curvature $\kappa$, compressibility $\sigma$, and the balance ratio $\Gamma =\kappa /\tau$. Structural strain compares an encoding to a designated reference representation, while compressibility measures reduction under canonical symbolic compression. These quantities are deterministic descriptors within the declared encoding class rather than coordinate-free invariants of the underlying operator. The structural length functional underlying these descriptors is developed, canonical compression is formalized, and finite symbolic comparison is distinguished from pathwise symbolic deformation. A robustness theorem shows that, away from the threshold surface $\Gamma =\sigma$, sufficiently small admissible perturbations preserve the induced diagnostic label. A supporting weight-robustness result further shows that qualitative labels persist across a local admissible family of weight choices under corresponding nondegeneracy conditions. The framework serves as a reproducible diagnostic for operator representations alongside Lyapunov, spectral, pseudospectral, and energy-based stability theories. Examples of representative ordinary and partial differential operators illustrate how the descriptors are computed and how they behave under admissible re-expression, while the appendices provide the technical backbone of the paper: formal definitions, reproducibility protocol, extended perturbation arguments, and explicit failure-mode analysis. Additional sensitivity checks regarding encoding, weights, and threshold variation clarify the method’s scope, and explicit failure modes delineate the boundary cases in which the descriptors cease to apply. The main contribution of this study is a formally delimited and reproducible symbolic framework for comparing differential operators under a fixed, declared specification, together with robustness results and worked examples that clarify the method’s scope. Full article

Unreinforced masonry walls exhibit limited resistance to lateral loads and, therefore, frequently require strengthening interventions. Carbon fiber reinforced polymer (CFRP) systems provide an efficient retrofit solution; however, current design procedures defined in structural guidelines require repetitive trial calculations to determine the necessary reinforcement amount. This study introduces a hybrid computational process that integrates metaheuristic optimization with symbolic regression to generate direct analytical equations for the estimation of the required CFRP area. First, a comprehensive database containing 1300 optimal strengthening scenarios was generated using the Jaya optimization algorithm under the constraints specified in ACI 440.7R and ACI 530. The resulting dataset was subsequently processed through symbolic regression using the PySR platform to identify explicit mathematical relationships between structural parameters and the optimum CFRP area. Most traditional machine learning approaches operate as black-box predictors. In contrast, the proposed approach generates interpretable closed-form expressions that can be used directly in engineering calculations. Two models were derived from the Pareto-optimal solution set. The first model is a simplified equation emphasizing algebraic simplicity. The second model prioritizes prediction accuracy. The simplified formulation achieved a coefficient of determination of approximately 0.992. The accuracy-focused model achieved a value above 0.997 with very low prediction errors. Validation studies with independent test samples showed that the obtained equations are reliable. The average error for the simplified model is below 4%, and for the high-accuracy model, it is approximately 2%. The results demonstrate that combining the optimization-generated datasets with symbolic regression makes it possible to obtain transparent design equations. These equations eliminate iterative design processes and provide a fast and reliable estimation tool for CFRP strengthening of masonry walls. Full article

We undertake a thorough examination of George Boole’s claim that, as he discovered by means of his algebra, the law of idempotence is “more fundamental” than the law of non-contradiction (The Laws of Thought, Chapter III, Proposition IV). There is a paucity of sources investigating this subject (with a notable exception being (Béziau 2018)). We query Boole’s claim; we examine if and how we can make sense of it; we identify the notable Aristotelian precedent of philosophical reflections on relative fundamentality of logical principles; and we inquire as to what philosophical view of logic is consistent with Boole’s way of thinking about logical principles. Boole’s thinking is apparently burdened by a metaphysically laden view of logic. We argue in detail that it is a radically different way of thinking about logic—a formalist view that regards logic as manipulation of symbolic resources, congenial to logical positivism—which allows us to make some tentative sense of claims about relative fundamentality of logical laws, insofar as we can define such a notion in a meaningful way. However, on the other hand, entanglements in metaphysically laden phantasmagorias fail to support (or perhaps even fail to make sense of) Boole’s claim. In order to substantiate the metalogical and philosophical–logical claims, we advance and construct formal derivations within different Boolean languages with a view to showing how idempotence is primary in some formal systems, but it is derivable (from non-contradiction) in other systems. Hence, Boole’s claim, as we can make sense of it (as relative derivability), is language-dependent, and we argue that this is consistent with a certain philosophical view of what logic is. Full article

Geometric problem-solving (GPS) has been a long-standing challenge in the fields of formal mathematics and artificial intelligence. To address the limitations of unidirectional approaches, we developed a neuro-symbolic system that integrates forward and backward reasoning. The neural component employs a gating-enhanced attention network to select candidate theorems, guiding the heuristic search and pruning irrelevant branches. The symbolic component is a bidirectional solver built on FormalGeo, which performs rigorous geometric relational reasoning and algebraic computation. The neural component predicts the theorems based on the current problem state, while the symbolic component applies these theorems and updates the problem state. These two parts interact iteratively until the problem is solved. The solving process is organized as a graph structure where facts and goals serve as nodes and theorems as edges, thereby generating a human-readable solution. The proposed neuro-symbolic system achieved an 89.63% problem-solving success rate (PSSR) on the FormalGeo7K dataset, surpassing the previous best result. Full article

A symbolic method is presented for examining the solvability and constructing the exact solution to boundary value problems for systems of linear ordinary loaded differential equations and loaded integro-differential equations with nonlocal boundary conditions. The method uses the inverse of the differential operator involved in the system of loaded differential or integro-differential equations. A solvability criterion based on the determinant of a matrix and an exact analytical matrix-form solution formula are presented. For the implementation of the method into computer algebra system software, two algorithms are provided. The effectiveness of the method is demonstrated by solving several problems. The theoretical and practical results obtained complement the existing literature on the subject. Full article

Cantor real numeration systems provide a natural algebraic source of self-similar aperiodic structures, extending the classical $\beta$-integers framework introduced in quasicrystal modeling by Gazeau. We study how the choice of algebraic parameters of the base $\mathcal{B}={\left({\beta }_i\right)}_{i\in \mathbb{Z}}$ influences the self-similarity and other combinatorial properties of the encoding symbolic sequence. These properties, namely repetitivity and palindromicity, are key features deciding the character of the spectrum of the underlying one-dimensional Schrödinger operator with aperiodic potential. Full article