Mathematics

We investigate a theory-informed attention framework combining Cognitive Load Theory (CLT)-inspired resource regulation with Token–Context Attention (TCA) and a TCA-specific application of root-mean-square (RMS) branch normalization. Seven primary conditions were evaluated across SNLI, SST-2, IMDb, AG News, and DBPedia 14 using 10 paired seeds and locked data manifests, with a validation-calibrated fixed-gate control for adaptive TCA-RMS. A separately locked secondary ablation added entropy-only CLT (CLT-E) to self-attention, TCA, and TCA-RMS under the same records and seeds. RMS branch normalization increased mean raw-TCA accuracy on all five datasets, with Holm-adjusted raw-TCA accuracy support on IMDb and AG News. In contrast, none of the prespecified direct mixed-CLT accuracy additions survived the within-dataset Holm correction; CLT-related evidence was localized to particular architectures, datasets, and metrics. No adaptive-versus-fixed accuracy contrast was detected after correction, and the entropy-only ablation shows that useful attenuation can arise without the full entropy–uncertainty–IDF composition. Accordingly, the results support branch-scale normalization and bounded attention-output gating as separable architectural interventions, while neither universal accuracy improvement nor human cognitive-load construct validity is claimed. These conclusions are restricted to lightweight, from-scratch classifiers trained under locked limited-data protocols; whether pretrained encoders change the need for TCA branch normalization or the effects of the CLT-inspired gate remains an open empirical question.

Quantum transition simulations on complex networks capture superposition, interference, and decoherence beyond classical stochastic models. This review provides a unified, mathematically rigorous account of the theory, algorithms, and cross-domain applications of quantum-transition-based network simulation. Starting from the Schrödinger equation and density matrix formalism, we develop the Lindblad master equation for open system dynamics and extend it to hybrid quantum–classical stochastic differential equations and hierarchical equations of motion for non-Markovian environments. Three quantum walk classes are formalized—continuous-time, discrete-time, and quantum stochastic walks—showing how decoherence and network topology govern the transition from ballistic quantum spreading to classical diffusion. Four simulation algorithms are compared: exact diagonalization [O(D⁴) memory, O(D⁶) diagonalization cost, D ≤ 20 states]; Krylov subspace [O(D²m), D∼100 states]; quantum trajectory Monte Carlo [O(DM), D∼10³ states]; and tensor networks [L∼10⁴ sites, where L is the number of sites and D = d^L is the Hilbert space dimension with local dimension d]. Seven metrics—fidelity, coherence, purity, participation ratio, mean first-passage time, von Neumann entropy, and spectral gap—are linked to experimental observables. Applications span physical systems (excitation transport), biological systems (environment-assisted quantum transport in the FMO complex), and socio-technical systems (opinion polarization). Key limitations and four emerging directions of machine learning-assisted optimization, non-Markovian hierarchical equations of motion, simulation-guided network design, and hybrid classical–quantum computation are identified.

A unified analytical relation between the generalized Rice–Cherepanov J-integral and Stroh’s complex potentials is established for anisotropic elastic and quasicrystalline materials. A symmetric representation of both components of the J-integral is first derived in terms of the displacement and stress-function vectors. This representation treats the two components on an equal footing and provides a direct component-wise connection with the Stroh complex potentials. Explicit relations between both components of the J-integral and the Stroh complex potentials are established, extending the analysis beyond a representation in terms of a single complex configurational-force quantity. The resulting complex representation is then extended to quasicrystalline solids by incorporating the coupled phonon and phason fields into the generalized Stroh formalism. The asymptotic behavior of the complex potentials near a mathematical cut is used to derive explicit expressions for both components of the J-integral in terms of generalized phonon–phason stress intensity factors. In addition to the conventional intensity factors associated with displacement jumps, the formulation incorporates factors associated with stress jumps across the discontinuity. These stress-jump factors are essential for describing discontinuities subject to additional constraints, such as partially or completely blocked phason displacements at the faces of a quasicrystalline crack. The first component of the J-integral is expressed through the Barnett–Lothe tensors and contains a positive-definite contribution from the conventional intensity factors and a negative-definite quadratic contribution associated with the stress-jump factors. Consequently, a nonzero stress-jump contribution, which generally arises when the phason field is partially or completely blocked on the faces of a quasicrystalline crack, can reduce the J-integral. The overall effect of the phason constraint also depends on a mixed term coupling the displacement-jump and stress-jump intensity factors. The second component is expressed through the Stroh matrices and the Barnett–Lothe tensors and retains the corresponding mixed and stress-jump contributions. The derived relations recover previously established results for classical cracks and one-dimensional quasicrystals as limiting cases. The formulation provides a unified analytical framework for configurational forces associated with cracks, deformable and rigid inclusions, and discontinuities with constrained phason fields.