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.
We define a distance analogous to the Gromov-Hausdorff distance that enables the comparison of arbitrary quasi-isometric spaces. We also investigate properties preserved under limits with respect to this distance, as well as properties of the entire class of metric spaces equipped with this distance. For this purpose, we introduce the notion of quasi-isometric distortion for correspondences. Using this notion, we prove that the class of all metric spaces is path-connected; in fact, any two metric spaces can be connected by a curve of finite length.
In this paper, for a given variety , we present a universal algorithm for constructing a subvariety of -dialgebras from which one can recover an algebra belonging to . Such a subvariety is called the variety of initial -dialgebras. In addition, we construct a basis of the free initial Lie and associative dialgebras.
Over the split-octonion algebra defined over an arbitrary field, we solve all polynomial equations whose coefficients are scalar except for the constant term. As an application, we determine the square and cubic roots of an octonion.
Recently, we have proposed a new free boundary problem representing the bread baking process in a hot oven. Unknown functions in this problem are the position of the evaporation front, the temperature field and the water content. For solving this problem we observed two difficulties that the growth rate of the free boundary depends on the water content and the boundary condition for the water content contains the temperature. In this paper, by improving the regularity of solutions, we overcome these difficulties and establish existence of a solution locally in time and its uniqueness. Moreover, under some sign conditions for initial data, we derive a result on the maximal interval of existence to solutions.
We survey recent results and current challenges concerning the growth rate inequality for sphere endomorphisms, and present a number of open problems and conjectures arising in this context.
In this survey we talk about what is known as Invariance Principle in dynamical systems. It states that the disintegration of measures with zero center Lyapunov exponents admits some extra invariance by holonomies. We focus on explaining the basic definitions and ideas behind a series of results about the Invariance Principle and give some basic applications on how this is used in dynamical systems.
Based on entropy and symmetrical uncertainty (SU), we define a metric for categorical random variables and show that this metric can be promoted into an appropriate quotient space of categorical random variables. Moreover, we also show that there is a natural commutative monoid structure in the same quotient space, which is compatible with the topology induced by the metric, in the sense that the monoid operation is continuous.
Our constructions provide a systematic way to study cohomology pre-algebraic structures via classical cohomology, simplifying computations and enabling the use of established techniques.
The purpose of this expository article is to give a down-to-hearth introduction to the notion of an arithmetic group and arithmetic manifold. To achieve this we have decided to bring two geometrical questions relating the growth of systole and kissing number in hyperbolic manifolds, as a motivational guide, whose answers so far are given by the use of arithmetic manifolds. As a consequence, we answer these questions in detail for dimension 2, we mention was is known for hyperbolic manifolds of higher dimension, and also for other locally symmetric spaces. We end the exposition with some open questions.
The Boltzmann equation has been a driving force behind significant mathematical research over the years. Its challenging theoretical complexity, combined with a wide variety of current scientific and technological problems that require numerical simulations based on this model, justifies such interest. This work provides a brief overview of studies and advances related to the solution of the linear Boltzmann equation in one- and two-dimensional spatial dimensions. In particular, relevant aspects of the discrete ordinates approximation of the model are highlighted for neutron and photon transport applications, including nuclear safeguards, nuclear reactor shielding problems, and optical tomography. In addition, a short discussion on rarefied gas dynamics problems, which are relevant, for instance, in the studies of micro-electro-mechanical systems, and their connection with the linearized Boltzmann equation, is presented. A primary goal of the work is to establish as much as possible the connections between the different phenomena described by the model and the versatility of the analytical methodology, the ADO method, in providing concise and accurate solutions, which are fundamental for numerical simulations.
We investigate restricted Lie algebras arising as analogues of (twisted) right-angled Artin groups and right-angled Coxeter groups over fields of characteristic two. These algebras are defined via quadratic relations determined by decorated graphs. We compute their cohomology rings with trivial coefficients and uncover phenomena specific to characteristic two: unlike in zero/odd characteristics, where quadratically defined ordinary and restricted Lie algebras have equivalent cohomology theories, the characteristic two case exhibits dependence on the base field. In particular, we prove that the ground field being the prime field $\mathbb F_2$ characterizes when a Lie-theoretic analogue of the twisted Droms theorem holds. Generalizations of graph Lie algebras are also discussed.
In this paper we provide an explicit tableaux realization for all simple subquotients of a relation Gelfand-Tsetlin $\mathfrak{gl}(n)$-module.
We revisit the notion of equations describing pseudospherical surfaces, starting from the works by Sasaki, whose roots were influenced by the AKNS system, the works by Chern and Tenenblat, until current research topics in the field relating to Cauchy problems involving these equations and their geometric consequences.
In this paper, we introduce the deformed homogeneous polynomials R_n(x,y;u|q). These polynomials generalize some classical polynomials: the Rogers-Szegö polynomials h_n(x|q), the generalized Rogers-Szegö polynomials r_n(x,y), the Stieltjes-Wigert polynomials S_n(x;q), among others. Basic properties of the polynomial R_n are given, along with recurrence relations, its q-difference equation, and representations. Generating functions for the polynomials R_n(x,y;u|q) are given. These functions include generalizations of the Mehler and Rogers formulas. In addition, generalizations of the q-binomial formula and the Heine transformation formula are obtained. These results are obtained via the u-deformed q-exponential operator E(yD_q|u), defined here. From this operator, we obtain for free the operators T(yD_q) the Chen, R(yD_q) of Saad, ℰ(yD_q) of Exton, and ℛ(yD_q) of Rogers-Ramanujan when u=1,q,√(q),q^2, respectively. We introduce the deformed basic hypergeometric series _rΦ_s, a generalization of the classical basic hypergeometric series. New transformation formulas for basic hypergeometric series are obtained.
We introduce noncommutative rings with DK-property (Dubrovin-Komarnytsky's property) and investigate elementary divisor rings with such property. Mostly we pay attention to these kinds of noncommutative rings which have stable range 1. A theory of reduction matrices over such rings is constructed. As a consequence, new families of non-commutative rings of elementary divisor rings are constructed.
This paper is devoted to the complete geometric classification of complex 5-dimensional solvable symmetric Leibniz algebras. As a corollary, we have the complete geometric classification of complex 5-dimensional symmetric Leibniz algebras.
