In this paper, we provide an explicit method for determining the zero set of monic quadratic general polynomials over Cayley–Dickson algebras with any base field of characteristic not equal to 2. For the special case of locally-complex Cayley–Dickson algebras over the reals, we prove that every such polynomial always has a root. We prove the existence of right eigenvalues for any 2 × 2 matrix over Cayley's Octonions (the real octonion division algebra), and our root-finding method allows the computation of some of the right eigenvalues.
In this article, we study quasi-isospectral operators as a generalization of isospectral operators. The paper contains both expository material and original results. We begin by reviewing known results on isospectral potentials on compact manifolds and finite intervals, and then introduce the notion of quasi-isospectrality. We next investigate the BMT method as a systematic approach to constructing quasi-isospectral Sturm-Liouville operators on a finite interval, and apply it to several boundary value problems. Our main result shows that any two quasi-isospectral closed manifolds of odd dimension are, in fact, isospectral. In addition, we extend classical compactness results for isospectral potentials on low-dimensional manifolds to the quasi-isospectral setting via heat trace asymptotics.
We explicitly find a complete set of ${1\over4}(n+2)^2$ (resp. ${1\over4}(n+1)(n+3)$) primitive orthogonal idempotents in ${\rm Sym}^n\mathbb{H}\otimes_\mathbb{R}\mathbb{C}$ if $n$ is even (resp. odd), where ${\rm Sym}^n\mathbb{H}$ is the $n^{\it th}$ symmetric power of the Hamilton quaternion algebra $\mathbb{H}$. We also give a complete set of ${1\over4}(n+2)^2$ (resp. ${1\over8}(n+1)(n+3)$) primitive orthogonal idempotents in ${\rm Sym}^n\mathbb{H}$ if $n$ is even (resp. odd). Moreover, we explicitly find a complete set of ${1\over24}(n+2)(n+3)(n+4)$ (resp. ${1\over24}(n+1)(n+3)(n+5)$) primitive orthogonal idempotents in the associative subalgebra $\big({\rm Sym}^n\mathbb{H}\cdot Z({\rm Sym}^n\mathbb{O})\big)\otimes_{\mathbb{R}}\mathbb{C}$ of ${\rm Sym}^n\mathbb{O}\otimes_\mathbb{R}\mathbb{C}$ if $n$ is even (resp. odd), where ${\rm Sym}^n\mathbb{O}$ is the $n^{\it th}$ symmetric power of the Cayley octonion algebra $\mathbb{O}$ and $Z({\rm Sym}^n\mathbb{O})$ is its center. We also give a complete set of ${1\over24}(n+2)(n+3)(n+4)$ (resp. ${1\over48}(n+1)(n+3)(n+5)$) primitive orthogonal idempotents in the associative subalgebra ${\rm Sym}^n\mathbb{H}\cdot Z({\rm Sym}^n\mathbb{O})$ of ${\rm Sym}^n\mathbb{O}$ if $n$ is even (resp. odd).
In this paper, we provide a comprehensive study of the structural properties of the transposed Poisson algebra 𝒲(a,-1). We classify several types of linear maps, including derivations, local derivations, quasi-derivations, and δ-derivations, showing that non-trivial δ-derivations exist only for δ=1 and δ=1/2. Furthermore, we describe the groups of automorphisms, local automorphisms, 2-local automorphisms, and quasi-automorphisms. We also investigate Rota–Baxter operators of weight 1 on 𝒲(a,-1). Specifically, we classify operators that are homogeneous with respect to both the standard 𝐙-grading and a 𝐙_2-grading, establishing a rigidity result for the latter case. Finally, we classify all 𝐖-compatible Novikov–Poisson structures, demonstrating that the associative product on the Witt algebra is universally compatible with its known Novikov structures.
In complex production lines, it is essential to have strict, fast-acting rules to determine whether the system is In Control (InC) or Out of Control (OutC). This study explores a bio-inspired method that digitally mimics ant colony behavior to classify InC/OutC states and forecast imminent transitions requiring maintenance. A case study on industrial potato chip frying provides the application context. During each two-minute frying cycle, sequences of eight temperature readings are collected. Each sequence is treated as a digital ant depositing virtual pheromones, generating a Base Score. New sequences, representing new ants, can either reinforce or weaken this score, leading to a Modified Base Score that reflects the system's evolving condition. Signals such as extreme temperatures, large variations within a sequence, or the detection of change-points contribute to a Threat Score, which is added to the Modified Base Score. Since pheromones naturally decay over time unless reinforced, an Environmental Score is incorporated to reflect recent system dynamics, imitating real ant behavior. This score is calculated from the Modified Base Scores collected over the past hour. The resulting Total Score, obtained as the sum of the Modified Base Score, Threat Score, and Environmental Score, is used as the main indicator for real-time system classification and forecasting of transitions from InC to OutC. This ant colony optimization-inspired approach provides an adaptive and interpretable framework for process monitoring and predictive maintenance in industrial environments.
