Axioms, Volume 13, Issue 12 (December 2024) – 44 articles

The aim of the article is to create a method for studying the asymptotics of solutions to second-order differential equations with irregular singularities. The method allows us to prove the convergence of formal series included in the asymptotics of solutions for a wide class of second-order differential equations in the neighborhoods of their irregular singular points, including the point at infinity, which is generally irregular. The article provides a number of applications of the method for studying the asymptotics of solutions to both ordinary differential equations and partial differential equations. Full article

This paper reports the research work carried out with the goal of geometrically and algebraically describing, as well as topologically classifying, the curves resulting from the intersection of a torus with families of parabolic and elliptical cylinders within a purely Euclidean framework. The parabolic cylinders under analysis have generatrices parallel to the axis of the torus, whereas the elliptical cylinders, centered at the same point as the torus, have axes either aligned with or orthogonal to the torus’s axis. For the topological classification of these intersection curves, we consider their number of connected components and self-intersection points. GeoGebra, which was used to create the 3D visual geometric representations of the intersection curves, and Maple, which was used to perform the essential symbolic algebraic calculations, were critical computational tools in the development of this work. Theoretical and computational approaches are interwoven throughout this study, with the computational work serving as the foundation for exploration and providing insights that contributed to the theoretical validation of the results revealed through GeoGebra simulations. Full article

Motivated by recent advancements in the quantum detection problem employing both discrete and continuous frames, this paper delves into a quantum detection problem utilizing g-frames within the context of quaternionic Hilbert spaces. We offer several equivalent representations of injective g-frames in separable quaternionic Hilbert spaces. By normalizing the trace, we establish a classification for the g-frame injectivity problem. Additionally, we propose a method to derive an injective g-frame by leveraging an injective frame within quaternionic Hilbert spaces. Furthermore, we demonstrate that the injectivity of a g-frame remains intact under a linear isomorphism, while injective g-frames exhibit instability in infinite-dimensional scenarios. Full article

In this article, we examine and investigate various variants of Julia set patterns for complex exponential functions $W\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+log{c}^t,$ and $T\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+\gamma$ (which are analytic except at $z=0$) where $n\ge 2,$ $m,n\in \mathbb{N},$ $\alpha ,\beta ,\gamma \in \mathbb{C},c\in \mathbb{C}\setminus \left\{0\right\}$ and $t\in \mathbb{R},t\ge 1,$ by employing a viscosity approximation-type iterative method. We employ the proposed iterative method to establish an escape criterion for visualizing Julia sets. We provide graphical illustrations of Julia sets that emphasize their sensitivity to different iteration parameters. We present graphical illustrations of Julia sets; the color, size, and shape of the images change with variations in the iteration parameters. With fixed input parameters, we observe the intriguing behavior of Julia sets for different values of n and m. We hope that the conclusions of this study will inspire researchers with an interest in fractal geometry. Full article

Our study presents an innovative variational Bayesian parameter estimation method for the Quantile Nonlinear Dynamic Latent Variable Model (QNDLVM), particularly when dealing with missing data and nonparametric priors. This method addresses the computational inefficiencies associated with the traditional Markov chain Monte Carlo (MCMC) approach, which struggles with large datasets and high-dimensional parameters due to its prolonged computation times, slow convergence, and substantial memory consumption. By harnessing the deterministic variational Bayesian framework, we convert the complex parameter estimation into a more manageable deterministic optimization problem. This is achieved by leveraging the hierarchical structure of the QNDLVM and the principle of efficiently optimizing approximate posterior distributions within the variational Bayesian framework. We further optimize the evidence lower bound using the coordinate ascent algorithm. To specify propensity scores for missing data manifestations and covariates, we adopt logistic and probit models, respectively, with conditionally conjugate mean field variational Bayes for logistic models. Additionally, we utilize Bayesian local influence to analyze the Ecological Momentary Assessment (EMA) dataset. Our results highlight the variational Bayesian approach’s notable accuracy and its ability to significantly alleviate computational demands, as demonstrated through simulation studies and practical applications. Full article

The efficiency and quality of the manufacturing industry are greatly influenced by production scheduling, which makes it a crucial aspect. A well-designed production scheduling scheme can significantly enhance manufacturing efficiency and reduce enterprise costs. This paper presents a tailored optimization model designed to address a more complex production scheduling problem that incorporates parallel machines and preventive maintenance. The proposed solutions aim to achieve a balance between job sequence and machine reliability, considering the minimum maintenance cost rate for determining maintenance cycles of deteriorating machines in real manufacturing scenarios. Furthermore, the objective of minimizing the maximum completion time guides machine assignment and job sequence based on maintenance constraints. The innovation lies in the introduction of a greedy algorithm that utilizes a water injection model to address this NP-hard integrated problem. A pre-distribution model is constructed using the water injection model, and its solution is utilized as input for constructing the production scheduling model, which aids in determining machine assignment and job sequence. This algorithm demonstrates remarkable effectiveness and efficiency, enabling the achievement of an optimal solution. A numerical example is presented to illustrate the computational process, accompanied by an extensive discussion of the results showcasing improved performance. Furthermore, the optimization model developed in this paper can be adapted to tackle the production scheduling problem with modifications tailored for parallel machines. Full article

The goal of this study is to derive new conditions that improve the testing of the oscillatory and asymptotic features of fourth-order differential equations with an advanced neutral term. By using Riccati techniques and comparison with lower-order equations, we establish new criteria that verify the absence of positive solutions and, consequently, the oscillation of all solutions to the investigated equation. Using our results to analyze a few specific instances of the examined equation, we can ultimately clarify the significance of the new inequalities. Our results are an extension of previous results that considered equations with a neutral delay term and also an improvement of previous results that considered only equations with an advanced neutral term. Full article

This manuscript presents set-theoretical solutions to the Yang–Baxter equation within the framework of GE-algebras by constructing mappings that satisfy the braid condition and exploring the algebraic properties of GE-algebras. Detailed proofs and the use of left and right translation operators are provided to analyze these algebraic interactions, while an algorithm is introduced to automate the verification process, facilitating broader applications in quantum mechanics and mathematical physics. Additionally, the Yang–Baxter equation is applied to spin transformations in quantum mechanical spin-${\displaystyle \frac{1}{2}}$ systems, with transformations like rotations and reflections modeled using GE-algebras. A Cayley table is used to represent the algebraic structure of these transformations, and the proposed algorithm ensures that these solutions are consistent with the Yang–Baxter equation, offering new insights into the role of GE-algebras in quantum spin systems. Full article

Understanding shear flow behavior in compressible, viscous, micropolar real gases is essential for both theoretical advancements and practical engineering applications. This study develops a comprehensive model that integrates micropolar fluid theory with compressible flow dynamics to accurately describe the behavior of real gases under shear stress. We formulate the governing equations by incorporating viscosity and micropolar effects and transform the obtained system into the mass Lagrangian coordinates. Two numerical methods, Faedo–Galerkin approximation and finite-difference methods, are used to solve it. These methods are compared using several benchmark examples to assess their accuracy and computational efficiency. Both methods demonstrate good performance, achieving equally precise results in capturing essential flow characteristics. However, the finite-difference method offers advantages in speed, stability, and lower computational demands. This research bridges gaps in existing models and establishes a foundation for further investigations into complex flow phenomena in micropolar real gases. Full article

The study of the Hankel determinant generated by the Maclaurin series of holomorphic functions belonging to particular classes of normalized univalent functions is one of the most significant problems in geometric function theory. Our goal in this study is first to define a family of alpha-convex functions associated with modified sigmoid functions and then to investigate sharp bounds of initial coefficients, Fekete-Szegö inequality, and second-order Hankel determinants. Moreover, we also examine the logarithmic and inverse coefficients of functions within a defined family regarding recent issues. All of the estimations that were found are sharp. Full article

This study computationally examined the Richtmyer–Meshkov instability (RMI) evolution in a helium backward-triangular bubble immersed in monatomic argon, diatomic nitrogen, and polyatomic methane under planar shock wave interactions. Using high-fidelity numerical simulations based on the compressible Navier–Fourier equations based on the Boltzmann–Curtiss kinetic framework and simulated via a modal discontinuous Galerkin scheme, we analyze the complex interplay of shock-bubble dynamics. Key findings reveal distinct thermal non-equilibrium effects, vorticity generation, enstrophy evolution, kinetic energy dissipation, and interface deformation across gases. Methane, with its molecular complexity and higher viscosity, exhibits the highest levels of vorticity production, enstrophy, and kinetic energy, leading to pronounced Kelvin–Helmholtz instabilities and enhanced mixing. Conversely, argon, due to its simpler atomic structure, shows weaker deformation and mixing. Thermal non-equilibrium effects, quantified by the Rayleigh–Onsager dissipation function, are most significant in methane, indicating delayed energy relaxation and intense turbulence. This study highlights the pivotal role of molecular properties, specific heat ratio, and bulk viscosity in shaping RMI dynamics in polyatomic gases, offering insights on uses such as high-speed aerodynamics, inertial confinement fusion, and supersonic mixing. Full article

This article investigates the applications of the q-Carlson–Shaffer operator on subclasses of q-uniformly starlike functions, introducing the class $S{T}_q(m,c,d,\beta )$. The study establishes a necessary condition for membership in this class and examines its behavior within conic domains. The article delves into properties such as coefficient bounds, the Fekete–Szegö inequality, and criteria defined via the Hadamard product, providing both necessary and sufficient conditions for these properties. Full article

The objective of this research article is to introduce Kikkawa and Suzuki-type contractions in the setting of topological vector space-valued cone metric space with a solid cone and establish some new fixed point results for multi-valued mappings. The problem of finding fixed points for multi-valued mappings satisfying locally contractive conditions on a closed ball is also addressed. Our findings generalize a number of well-established results in the literature. To highlight the uniqueness of our key finding, we present an example. As a demonstration of the applicability of our principal theorem, we prove a result in homotopy theory. Full article

The field of indices has been explored and advanced by various researchers for different purposes. One purpose is the optimization of indices in various problems. In this work, the general power-sum connectivity index is considered. The general power-sum connectivity index was investigated for k-generalized quasi-trees where optimal graphs were found. Further, in this work, we extend the idea of optimization to families of graphs, including uni-cyclic graphs, bi-cyclic graphs and trees. The optimization is carried out by means of operations named as Operation A, B, C and D. The first two operations increase the value of the general power-sum connectivity index, while the last two work opposite to Operations A and B. These operations are explained by means of diagrams, where one can easily obtain their working procedures. Full article

In this paper, we consider various aspects of the Schur problem for a square complex matrix A, namely the similarity unitary transformation of A into upper triangular form containing the eigenvalues of A on its diagonal. Since the profound work of I. Schur published in 1909, this has become a fundamental issue in the theory and applications of matrices. Nevertheless, certain details concerning the Schur problem need further clarification, especially in connection with the perturbation analysis of the Schur decomposition relative to perturbations in the matrix A. We consider both canonical and condensed Schur forms. Special attention is paid to matrices with simple eigenvalues. Some new concepts, such as quasi-Schur forms and diagonally spectral matrices, are also introduced and studied. Full article

Quantitative bisimulations between weighted finite automata are defined as solutions of certain systems of matrix-vector inequalities and equations. In the context of fuzzy automata and max-plus automata, testing the existence of bisimulations and their computing are performed through a sequence of matrices that is built member by member, whereby the next member of the sequence is obtained by solving a particular system of linear matrix-vector inequalities and equations in which the previously computed member appears. By modifying the systems that define bisimulations, systems of matrix-vector inequalities and equations with k unknowns are obtained. Solutions of such systems, in the case of existence, witness to the existence of a certain type of partial equivalence, where it is not required that the word functions computed by two WFAs match on all input words, but only on all input words whose lengths do not exceed k. Solutions of these new systems represent finite sequences of matrices which, in the context of fuzzy automata and max-plus automata, are also computed sequentially, member by member. Here we deal with those systems in the context of WFAs over the field of real numbers and propose a different approach, where all members of the sequence are computed simultaneously. More precisely, we apply a simultaneous approach in solving the corresponding systems of matrix-vector equations with two unknowns. Zeroing neural network (ZNN) neuro-dynamical systems for approximating solutions of heterotypic bisimulations are proposed. Numerical simulations are performed for various random initial states and comparison with the Matlab, linear programming solver linprog, and the pseudoinverse solution generated by the standard function pinv is given. Full article

In this paper, we introduce a novel class of graphs referred to as the Horadam–Lucas cubes. This class extends the concept of Lucas cubes and retains numerous desirable properties associated with them. Horadam–Lucas cubes can also be viewed as a companion graph family of the Horadam cubes, in a similar way the Lucas cubes relate to Fibonacci cubes or the Lucas-run graphs relate to Fibonacci-run graphs. As special cases, they also give rise to new graph families, such as Pell–Lucas cubes and Jacobsthal–Lucas cubes. We derive the several metric and enumerative properties of these cubes, including their diameter, periphery, radius, fundamental decomposition, number of edges, cube polynomials, and generating function of the cube polynomials. Full article

The decision tree algorithm is widely used in various classification problems due to its ease of implementation and strong interpretability. However, information in the real world often has uncertainty and partial reliability, which poses challenges for classification tasks. To address this issue, this paper proposes a fuzzy decision tree based on fuzzy rough sets and Z-numbers, aimed at enhancing the decision tree’s ability to handle fuzzy and uncertain information. In the aspect of rule extraction, we combine the fuzzy rough set model to propose a fuzzy confidence based on lower approximation as a metric for attribute selection, effectively addressing the role of imprecise knowledge in classification. In terms of the tree structure, the concept of Z-numbers is introduced, specifically focusing on the fuzzy constraint reliability B, making the information representation more aligned with human evaluation habits, as well as using Z-number rules to replace traditional fuzzy rules in constructing the fuzzy decision tree. Furthermore, as generating Z-numbers still presents certain challenges, this paper also establishes a method for reasonably generating Z-numbers in situations with limited information, utilizing the generated fuzzy constraint reliability B to adjust fuzzy numbers A. Finally, the proposed decision tree algorithm is experimentally compared with other classifiers, and the results indicate that this algorithm demonstrates higher classification accuracy and a more concise tree structure when handling datasets containing fuzzy and uncertain factors. This research enriches the existing research on fuzzy decision trees and shows greater potential in solving practical problems. Full article

We study the characteristics of affine vector fields on pseudo-Riemannian manifolds and provide tensorial formulas that characterize these vector fields. Using our approach, we present a straightforward proof that any affine vector field on a compact Riemannian manifold is a Killing vector field. Moreover, we establish several necessary and sufficient conditions for an affine vector field on a Riemannian manifold to be classified as a Killing or parallel vector field. Full article

In this paper, we study the existence and uniqueness of second-order impulsive delay differential systems. Firstly, we define cosine-type and sine-type delay matrix functions, which are used to derive the solutions of the impulsive delay differential systems. Secondly, based on the Schauder and Banach fixed-point theorems, we establish sufficient conditions that guarantee the existence and uniqueness of solutions to nonlinear impulsive delay differential systems. Finally, several examples are given to illustrate our theoretical results. Full article

In this article, we present a novel B-Polynomial Approach for approximating solutions to partial differential equations (PDEs), addressing the multiple initial conditions. Our method stands out by utilizing two-dimensional Bernstein polynomials (B-polynomials) in conjunction with their operational matrices to effectively manage the complexity associated with PDEs. This approach not only enhances the accuracy of solutions but also provides a structured framework for tackling various boundary conditions. The PDE is transformed into a system of algebraic equations, which are then solved to approximate the PDE solution. The process is divided into two main steps: First, the PDE is integrated to incorporate all initial and boundary conditions. Second, we express the approximate solution using B-polynomials and determine the unknown expansion coefficients via the Galerkin finite element method. The accuracy of the solution is assessed by adjusting the number of B-polynomials used in the expansion. The absolute error is estimated by comparing the exact and semi-numerical solutions. We apply this method to several examples, presenting results in tables and visualizing them with graphs. The approach demonstrates improved accuracy as the number of B-polynomials increases, with CPU time increasing linearly. Additionally, we compare our results with other methods, highlighting that our approach is both simple and effective for solving multidimensional PDEs imposed with multiple initial and boundary conditions. Full article

The encapsulation of particular quality functions and predicates within temporal logic formulas markedly enhances the representation of detailed temporal characteristics within a system. During our preliminary investigations, we innovatively combined quality constraint functions and predicates with Possibility Linear Temporal Logic (PoLTL), yielding the conception of Fuzzy Linear Temporal Logic with Quality Constraints (QFLTL). This amalgamation results in a significant elevation of QFLTL’s expressivity relative to PoLTL, ensuring the preservation of informational integrity whilst achieving a synchronized, yet selectively inclined, and exact consolidation of path reachability specifics alongside property satisfaction evaluations. This treatise represents a significant contribution to the field by integrating quality constraint functions and predicates into Possibility Computation Tree Temporal Logic (PoCTL), thus giving rise to Fuzzy Computation Tree Temporal Logic with Quality Constraints (QFCTL). We provide a comprehensive definition of QFCTL’s syntax, conduct an in-depth analysis of its logical characteristics, outline a precise model checking algorithm for QFCTL, and perform a meticulous complexity assessment of said algorithm. It is illustrated by examples that PoCTL is a proper subset of QFCTL, and QFCTL has stronger expressive power than PoCTL and can characterize more refined temporal properties of the system. An in-depth exploration of the logical characteristics of QFCTL was carried out, showing its unique logical characteristics that are distinct from other temporal logic systems under the influence of quality constraints. In particular, the introduction of characteristic predicates effectively classifies the satisfaction of temporal formulas, making the logical framework of QFCTL more complete compared to the existing probabilistic temporal logic. Moreover, by enriching QFCTL with a quantitative characteristic predicate operator, we innovate, culminating in the development of an enhanced Fuzzy Computation Tree Temporal Logic with Quality Constraints (QFCTL*). The logical characteristics of QFCTL* are explored in detail. It is shown that with the support of quantitative feature predicates, QFCTL* can divide the satisfaction of temporal formulas more delicately than QFCTL. The decision theorems for the semantics of QFCTL* formulas containing quantitative feature predicates are given, and the decidability of QFCTL* is strictly proved. Through the bounded-depth search of GPKS, a model-checking algorithm of QFCTL* on GPKS is presented. The correctness of the algorithm is proved, and the complexity of the algorithm is analyzed. In order to prove the practical applications and strong expressive capabilities of QFCTL and QFCTL*, we present a model-checking example as empirical evidence for the effectiveness of the proposed model-checking algorithms. Through this example, we verify that, compared with the existing PoCTL, QFCTL and QFCTL* can avoid the loss of system path reachability information or system property satisfaction information, ensure the synchronization of the two types of information, and fuse these two types of information according to weight preferences. QFCTL and QFCTL* can also synthesize temporal formulas that characterize the subproperties of the system according to weight preferences. These application examples also verify that the QFCTL and QFCTL* model-checking algorithms proposed in this article are automatic and effective. Full article

In this study, we propose a smoothed weighted quantile regression (SWQR), which combines convolution smoothing with a weighted framework to address the limitations. By smoothing the non-differentiable quantile regression loss function, SWQR can improve computational efficiency and allow for more stable model estimation in complex datasets. We construct an efficient optimization process based on gradient-based algorithms by introducing weight refinement and iterative parameter estimation methods to minimize the smoothed weighted quantile regression loss function. In the simulation studies, we compare the proposed method with two existing methods, including martingale-based quantile regression (MartingaleQR) and weighted quantile regression (WeightedQR). The results emphasize the superior computational efficiency of SWQR, outperforming other methods, particularly WeightedQR, by requiring significantly less runtime, especially in settings with large sample sizes. Additionally, SWQR maintains robust performance, achieving competitive accuracy and handling the challenges of right censoring effectively, particularly at higher quantiles. We further illustrate the proposed method using a real dataset on primary biliary cirrhosis, where it exhibits stable coefficient estimates and robust performance across quantile levels with different censoring rates. These findings highlight the potential of SWQR as a flexible and robust method for analyzing censored data in survival analysis, particularly in scenarios where computational efficiency is a key concern. Full article

In this paper, we search for some approaches for generating (1+1)-dimensional, (2+1)-dimensional and (3+1)-dimensional integrable equations by making use of various Lie algebras and the corresponding loop algebras under the frame of the Tu scheme. The well-known modified KdV equation, the heat conduction equation, the nonlinear Schrödinger equation, the (2+1)-dimensional cylindrical dissipative Zaboloskaya–Khokhlov equation and the (3+1)-dimensional heavenly equation are obtained, respectively. In addition, some new isospectral integrable hierarchies and their nonisospectral integrable hierarchies are singled out. All the Lie algebras and their loop algebras presented in the paper can be extensively applied to investigate other integrable hierarchies of evolution equations. Full article

In this paper, we study the existence of fixed points for interpolative Reich–Rus–Ćirić-type contractions in the setting of rectangular m-metric spaces. The use of the rectangular inequality, in place of the conventional triangle inequality, introduces a higher level of complexity in the computations, requiring more careful and refined analysis. We consider two distinct cases based on the sum of the interpolative exponents: one where the sum is less than 1, and another where the sum exceeds 1. The results we present generalize several existing theorems in the literature, and each is supplemented with illustrative examples to demonstrate their applicability. Moreover, we deduce corresponding results on m-metric spaces from these results, which are new themselves. They are also validated through examples. Full article

The Aharonov–Bohm effect can be described as a phase difference in interfering charged particles that travel through two distinct pathways oppositely surrounding a perpendicularly-positioned solenoid. The magnetic field emanates from the solenoid but does not intersect the pathways. On the other hand, the Kuramoto model can be used to identify the synchronization conditions that lead to a particular phase difference by treating the phases as coupled oscillators. Starting with the overall wave function expression for the electron in an Aharonov–Bohm potential, we derive a version of the Kuramoto model describing the phase dynamics of the bound state of the quantum mechanical system. We show that the resulting synchronization condition of the model coincides with the allowable values of the flux parameter for our case to achieve an Aharonov–Bohm effect. Full article

An even ring system G is a simple 2-connected plane graph with all interior vertices of degree 3, all exterior vertices of either degree 2 or 3, and all finite faces of an even length. G is angularly connected if all of the peripheral segments of G have odd lengths. In this paper, we show that every angularly connected even ring system G, which does not contain any triple of altogether-adjacent peripheral faces, has a perfect matching. This was achieved by finding an appropriate edge coloring of G, derived from the proof of the existence of a proper face 3-coloring of the graph. Additionally, an infinite family of graphs that are face 3-colorable has been identified. When interpreted in the context of the inner dual of G, this leads to the introduction of 3-colorable graphs containing cycles of lengths 4 and 6, which is a supplementation of some already known results. Finally, we have investigated the concept of the Clar structure and Clar set within the aforementioned family of graphs. We found that a Clar set of an angularly connected even ring system cannot in general be obtained by minimizing the cardinality of the set A. This result is in contrast to the previously known case for the subfamily of benzenoid systems, which admit a face 3-coloring. Our results open up avenues for further research into the properties of Clar and Fries sets of angularly connected even ring systems. Full article

In this study, the connection between certain subfamilies of harmonic univalent functions is established by utilizing a convolution operator involving the Mittag–Leffler function. The investigation reveals inclusion relations concerning harmonic $\gamma$-uniformly starlike mappings in the open unit disc, harmonic starlike functions and harmonic convex functions, highlighting the improvements given by the results presented here on previously published outcomes. Full article

In 1812, Gauss stated the following identity: ${\displaystyle {}_{2}F_1(a,b;c;1)=\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }(c-a-b)}{\mathrm{\Gamma }(c-a)\mathrm{\Gamma }(c-b)}}$, where, in the real case, $c-a-b>0$ and as an immediat consequence the Chu–Vandermonde identity: ${\displaystyle {}_{2}F_1(a,-n;c;1)=\frac{{(c-a)}_n}{{\left(c\right)}_n}}$ for any positive integer n. In this paper, we investigate the case when $c-a-b<0$ by taking $c=2b=-2n$, n and a are positive integers ($c-a-b=-n-a<0$). We give two significant applications stemming from these findings. The second part of the paper will be devoted to Kummer’s conditions concerning hypergeometric quadratic transformations, particularly focusing on the distinctions between the conditions provided by Gradshteyn and Ryzhik (GR) and those by Erdélyi, Magnus, Oberhettinger, and Tricomi (EMOI) are outlined. We establish that the conditions given by GR differ from those of EMOI, and we explore the methodologies employed by both groups in deriving their results. This leads us to conclude that the search for exact and unified conditions remains an open problem. Full article

We investigate from an algebraic and topological point of view the minimal prime spectrum of a universal algebra, considering the prime congruences with respect to the term condition commutator. Then we use the topological structure of the minimal prime spectrum to study extensions of universal algebras that generalize certain types of ring extensions. Our results hold for semiprime members of semidegenerate congruence–modular varieties, as well as semiprime algebras whose term condition commutators are commutative and distributive with respect to arbitrary joins and satisfy certain conditions on compact congruences, even if those algebras do not generate congruence–modular varieties. Full article