This article focuses on the classification, α-inner-derivations and α-centroids of complex Hom-trialgebras up to dimension three. The initial research on these algebras was conducted by Loday and Ronco, and this paper builds upon their work by utilizing computer algebra software (Mathematica) to analyze the equations that define the structure constants. Furthermore, we explore the concept of α-inner-derivations and α-centroids of complex Hom-trialgebras. The findings reveal that for 2- and 3-dimensional algebras, there is only one trivial α-inner-derivation. However, there exist 23 non-isomorphic α-inner-derivations for 2- and 3-dimensional algebras. Regarding α-centroids, we identify trivial isomorphism classes for 2- and 3-dimensional Hom-trialgebras. Additionally, there are 11 non-isomorphic classes for 2-dimensional Hom-trialgebras and 19 for 3-dimensional algebras. The range of dimensions for both α-inner-derivations and α-centroids spans from 0 to 3.

We establish a separation principle for a class of fractional order time-delay nonlinear differential systems. We show that a nonlinear time-delay observer is globally convergent and give sufficient conditions under which the observer-based controller stabilises the system.

We investigate the first cohomology space associated with the embedding of the Lie Orthosymplectic superalgebra $\mathfrak{osp}(n|2)$ on the (1,n)-dimensional superspace ℝ^1|n in the Lie superalgebra $ \mathcal{S}\Psi\mathcal{DO}(n)$ of superpseudodifferential operators with smooth coefficients, where n = 0, 1, 2. Following Ovsienko and Roger, we give erxplicit expressions of the basis cocycles.

We compute the second 𝔬𝔰𝔭(1 | 2) −relative cohomology space of 𝒦(1) with coefficients in the module of λ-densities 𝔉_λ on S ^1|1. This result allows us to compute the second 𝔬𝔰𝔭(1 | 2) −relative cohomology space of 𝒦(1) with coefficients in the Poisson superalgebra 𝒮𝒫. We explicitly give 2-cocycles spanning these cohomology spaces.