Symmetry

This paper establishes a comprehensive analytical framework for a new transformation of probability measures, denoted by , which unifies the classical t- and $T_a$-transformations in free probability. We derive the functional equation characterizing through the Cauchy–Stieltjes transform and explicitly show how it specializes to known deformations when $a=0$ or $t=1$. Within the setting of Cauchy-Stieltjes kernel families, we prove structural symmetry and invariance properties of the transformation, demonstrating in particular that both the free Meixner family and the free analog of the Letac-Mora class remain invariant under . Furthermore, we obtain several new limiting theorems that uncover symmetric relationships among fundamental free distributions, including the semicircular, Marchenko–Pastur, and free binomial laws.

We introduce a BiHom-type skew-symmetric bracket on general linear Lie algebra built from two commuting inner automorphisms $\alpha =A{d}_{\psi }$ and $\beta =A{d}_{\varphi }$, with and integers $i,j$. We prove that is a BiHom–Lie algebra, and we study the Lax equation obtained by replacing the commutator in the finite nonperiodic Toda lattice by this bracket. For the symmetric choice $\varphi =\psi$ with , the deformed flow is equivariant under conjugation and becomes gauge-equivalent, via , to a Toda-type Lax equation with a conjugated triangular projection. In particular, scalar deformations amount to a constant rescaling of time. On embedded $2\times 2$ blocks, we derive explicit trigonometric and hyperbolic formulae that make symmetry constraints (e.g., tracelessness) transparent. In the asymmetric hyperbolic case, we exhibit a trace obstruction showing that the right-hand side is generically not a commutator, which amounts to symmetry breaking of the isospectral property. We further extend the construction to the weakly coupled Toda lattice with an indefinite metric and provide explicit $2\times 2$ solutions via an inverse-scattering calculation, clarifying and correcting certain formulas in the literature. The classical Toda dynamics are recovered at special parameter values.

The neural mechanisms of auditory and visual processing are not only a core research focus in cognitive neuroscience but also hold critical importance for the development of brain–computer interfaces, neurological disease diagnosis, and human–computer interaction technologies. However, EEG-based studies on classifying auditory and visual brain activities largely overlook the in-depth utilization of spatial distribution patterns and frequency-specific characteristics inherent in such activities. This paper proposes an analytical framework that constructs symmetrical spatio-temporal–frequency feature association vectors to represent brain activities by computing EEG microstates across multiple frequency bands and brain functional connectivity networks. Then we construct an Adaptive Tensor Fusion Network (ATFN) that leverages feature association vectors to recognize brain activities related to auditory, visual, and audiovisual processing. The ATFN includes a feature fusion and selection module based on differential feature enhancement, a feature encoding module enhanced with attention mechanisms, and a classifier based on a multilayer perceptron to achieve the efficient recognition of audiovisual brain activities. The feature association vectors are then processed by the Adaptive Tensor Fusion Network (ATFN) to efficiently recognize different types of audiovisual brain activities. The results show that the classification accuracy for auditory, visual, and audiovisual brain activity reaches 96.97% using the ATFN, demonstrating that the proposed symmetric spatio-temporal–frequency feature association vectors effectively characterize visual, auditory, and audiovisual brain activities. The symmetrical spatio-temporal–frequency feature association vectors establish a computable mapping that captures the intrinsic correlations among temporal, spatial, and frequency features, offering a more interpretable method to represent brain activities. The proposed ATFN provides an effective recognition framework for brain activity, with a potential application for brain–computer interfaces and neurological disease diagnosis.