Search Results (853)

This paper presents a novel four-dimensional autonomous conservative model characterized by an infinite set of equilibrium points and an unusual algebraic structure in which all eigenvalues of the Jacobian matrix are zero. The linearization of the proposed model implies that classical stability analysis is inadequate, as only the center manifolds are obtained. Consequently, the stability of the system is investigated through both analytical and numerical methods using Lyapunov functions and numerical simulations. The proposed model exhibits rich dynamics, including hyperchaotic behavior, which is characterized using the Lyapunov exponents, bifurcation diagrams, sensitivity analysis, attractor projections, and Poincaré map. Moreover, in this paper, we explore the model with fractional-order derivatives, demonstrating that the fractional dynamics fundamentally change the geometrical structure of the attractors and significantly change the system stability. The Grünwald–Letnikov formulation is used for modeling, while numerical integration is performed using the Caputo operator to capture the memory effects inherent in fractional models. Finally, an analog electronic circuit realization is provided to experimentally validate the theoretical and numerical findings. Full article

Conjecture $\mathcal{O}$ (and the Gamma Conjectures), introduced by Galkin, Golyshev, and Iritani stand as pivotal open problems in the quantum cohomology of Fano manifolds, bridging algebraic geometry, mathematical physics, and representation theory. These conjectures aim to decode the structural essence of quantum multiplication by uncovering profound connections between spectral properties of quantum cohomology operators and the underlying geometry of Fano manifolds. Conjecture $\mathcal{O}$ specifically investigates the spectral simplicity and eigenvalue distribution of the operator associated with the first Chern class $c_1$ in quantum cohomology rings, positing that its eigenvalues govern the convergence and asymptotic behavior of quantum products. Full article

The Industrial Internet of Things (IIoT), a key enabler of Industry 4.0, integrates advanced communication technologies with the industrial economy to enable intelligent manufacturing and interconnected systems. Secure and reliable identity authentication in the IIoT becomes essential as connectivity expands across devices, systems, and domains. Blockchain technology presents a promising solution due to its decentralized, tamper-resistant, and traceable characteristics, facilitating secure and transparent identity verification. However, current blockchain-based cross-domain authentication schemes often lack a lightweight design, rendering them unsuitable for latency-sensitive and resource-constrained industrial environments. This paper proposes a lightweight cross-domain authentication scheme that combines blockchain with Chebyshev chaotic mapping. Unlike existing schemes relying heavily on Elliptic Curve Cryptography or bilinear pairing, our design circumvents such computationally intensive primitives entirely through the algebraic structure of Chebyshev polynomials. A formal security analysis using the Real-Or-Random (ROR) model demonstrates the scheme’s robustness. Furthermore, performance evaluations conducted with Hyperledger Fabric and the MIRACL cryptographic library validate the method’s effectiveness and superiority over existing approaches in terms of both security and operational efficiency. Full article

This paper is about a problem at the intersection of differential geometry, spectral analysis and the theory of manifolds. The study of finite-type subvarieties was initiated by Chen in the 1970s, with the aim of obtaining improved estimates for the mean total curvature of compact subvarieties in Euclidean space. The concept of a finite-type subvariety naturally extends that of a minimal subvariety or surface, the latter being closely related to variational calculus. In this work, we classify factorable surfaces in the Lorentz–Heisenberg space ${\mathbb{H}}_3$, equipped with a flat metric satisfying ${\Delta }^I{r}_i={\lambda }_i{r}_i$, which satisfies algebraic equations involving coordinate functions and the Laplacian operator with respect to the surface’s first fundamental form. Full article

Suppose X and Y are Banach spaces, K is a compact Hausdorff space, and $C(K, X)$ is the Banach space of all continuous X-valued functions (with the supremum norm). We will study some strongly bounded operators $T:C(K, X)\to Y$ with representing measures $m:\Sigma \to L(X,Y)$, where $L(X,Y)$ is the Banach space of all operators $T:X\to Y$ and $\Sigma$ is the $\sigma$-algebra of Borel subsets of K. The classes of operators that we will discuss are the Grothendieck, p-limited, p-compact, limited, operators with completely continuous, unconditionally converging, and p-converging adjoints, compact, and absolutely summing. We give a characterization of the limited operators (resp. operators with completely continuous, unconditionally converging, p-convergent adjoints) in terms of their representing measures. Full article

Uninorms are aggregation operators that generalize the t-norms (t-conorms), which are extensions of the logical connectives $\wedge (\vee )$ to the fuzzy set theory. The methods of constructing uninorms on more general algebraic structures (such as bounded posets, lattices, etc.) are an important subject of study, including an extensive work concerning these operations on the unit real interval $[0, 1]$. The construction of uninorms on bounded lattices has been extensively studied using various aggregation functions, such as t-norms, t-conorms, and t-subnorms. In this paper, we present construction methods for uninorms, based on the elements of a lattice, without using the existence of the mentioned operators. We determine the necessary and sufficient conditions for the introduced construction methods to result in the uninorms. Then, we show the differences between our methods and several methods known from the literature, including some illustrative examples. Full article

The objective of this study is to reveal the intrinsic connection between fractal operators in physical fractal spaces and continued fractions. The specific contributions include: (1) reviewing fundamental concepts of continued fractions and physical fractal theory; (2) establishing algebraic structure consistency between continued fractions and fractal operators through the medium of generation mappings; (3) discussing the convergence of fractal operators by employing theory from continued fraction analysis; and (4) confirming the correspondence between fixed points of infinite continued fractions and algebraic equations governing fractal operators. Full article

Aiming at the problem that the aggravation of the wheel tread wear of high-speed trains leads to the deterioration of train operation performance and an increase in re-profiling times, a multi-objective data-driven optimization design method for the wheel profile is proposed. Firstly, the chaotic map is introduced into the population initialization process of the golden jackal algorithm. In the later stage of the algorithm iteration, random disturbance is introduced with optimization algebra as the switching condition to obtain an improved optimization algorithm, and the performance index of the optimization algorithm is verified to be superior to other algorithms. Secondly, the improved multi-objective optimization algorithm and data-driven model are used to optimize the tread coordinates and obtain an optimized profile. The vehicle dynamics performance of the optimized profile and the wheel wear evolution after long-term service are compared. The results show that the tread wear index of the left and right wheels in a straight line is reduced by 62.4% and 62.6%, respectively, and the wear index of the left and right wheels in a curved line is reduced by 26.5% and 5.5%, respectively. The stability and curve passing performance of the optimized profile are improved. Under the long-term service conditions of the train, the wear amount of the optimized profile is greatly reduced. After the wear prediction of 200,000 km, the wear amount of the optimized profile is reduced by 60.1%, and it has better curve-passing performance. Full article

Using the Dixmier angle between two closed subspaces of a complex Hilbert space $\mathcal{H}$, we establish the necessary and sufficient conditions for the operator norm of the sum of two orthogonal projections, $P_{{\mathcal{W}}_1}$ and $P_{{\mathcal{W}}_2}$, onto closed subspaces ${\mathcal{W}}_1$ and ${\mathcal{W}}_2$, to attain its maximum, namely $\parallel P_{{\mathcal{W}}_1}+P_{{\mathcal{W}}_2}\parallel =2.$ These conditions are expressed in terms of the geometric relationship and symmetry between the ranges of the projections. We apply these results to orthogonal projections associated with a closed-range operator via its Moore–Penrose inverse. Additionally, for any bounded operator T with closed range in $\mathcal{H}$, we derive sufficient conditions ensuring $\parallel T{T}^{†}+T^{†}T\parallel =2,$ where $T^{†}$ denotes the Moore–Penrose inverse of T. This work highlights how symmetry between operator ranges and their algebraic structure governs norm extremality and extends a recent finite-dimensional result to the general Hilbert space setting. Full article

In this paper, a solution to the problem of electromagnetic field scattering on a periodic, constrained, planar antenna structure placed on the boundary of two dielectric media was formulated. The scattering matrix of such a structure was derived, and its generalization for the case of an antenna with a multilayer dielectric substrate was defined. By applying the Galerkin spectral method, the problem was reduced to a system of algebraic equations for the coefficients of current distribution on metal elements of the antenna grid, considering the distribution of the electromagnetic field on Floquet harmonics. The finite transverse dimension of the antenna was considered by introducing, to the solution of the case of an unconstrained antenna, a window function on the antenna aperture. The presented formalism allows modeling the operation of periodic, dielectric, composite antenna arrays. Full article

Darboux transformations are relations between the eigenfunctions and coefficients of a pair of linear differential operators, while Painlevé equations are nonlinear ordinary differential equations whose solutions arise in diverse areas of applied mathematics and mathematical physics. Here, we describe the use of confluent Darboux transformations for Schrödinger operators, and how they give rise to explicit Wronskian formulae for certain algebraic solutions of Painlevé equations. As a preliminary illustration, we briefly describe how the Yablonskii–Vorob’ev polynomials arise in this way, thus providing well-known expressions for the tau functions of the rational solutions of the Painlevé II equation. We then proceed to apply the method to obtain the main result, namely, a new Wronskian representation for the Ohyama polynomials, which correspond to the algebraic solutions of the Painlevé III equation of type $D_7$. Full article

Classical algebraic structures—such as magmas, groups, and Lie groups—are characterized by increasingly strong requirements in binary operation, ranging from no additional constraints to associativity, identity, inverses, and smooth-manifold structures. The hyperstructure paradigm extends these notions by allowing the operation to return subsets of elements, giving rise to hypermagmas, hypergroups, and Lie hypergroups, along with their variants such as quotient, reduced, and fuzzy hypergroups. In this work, we introduce the concept of superhyperstructures, obtained by iterating the powerset construction, and develop the theory of superhypermagmas and Lie superhypergroups. We further define and analyze quotient superhypergroups, reduced superhypergroups, and fuzzy superhypergroups, exploring their algebraic properties and interrelationships. Full article

This paper establishes some links between Sturm–Liouville problems and the well-known controllability property in linear dynamic systems, together with a control law design that allows any prefixed arbitrary final state finite value to be reached via feedback from any given finite initial conditions. The scheduled second-order dynamic systems are equivalent to the stated second-order differential equations, and they are used for analysis purposes. In the first study, a control law is synthesized for a forced time-invariant nominal version of the current time-varying one so that their respective two-point boundary values are coincident. Afterward, the parameter that fixes the set of eigenvalues of the Sturm–Liouville system is replaced by a time-varying parameter that is a control function to be synthesized without performing, in this case, any comparison with a nominal time-invariant version of the system. Such a control law is designed in such a way that, for given arbitrary and finite initial conditions of the differential system, prescribed final conditions along a time interval of finite length are matched by the state trajectory solution. As a result, the solution of the dynamic system, and thus that of its differential equation counterpart, is subject to prefixed two-point boundary values at the initial and at the final time instants of the time interval of finite length under study. Also, some algebraic constraints between the eigenvalues of the Sturm–Liouville system and their evolution operators are formulated later on. Those constraints are based on the fact that the solutions corresponding to each of the eigenvalues match the same two-point boundary values. Full article

This article proposes a method to solve nonlinear optimal control problems with arbitrary performance indices and terminal constraints, which is based on the neighboring extremal method and Gauss pseudospectral collocation. Firstly, a quadratic performance index is formulated, which minimizes the second-order variation of the nonlinear performance index and fully considers the deviations in initial states and terminal constraints. Secondly, the first-order necessary conditions are applied to derive the perturbation differential equations involving deviations in state and costate variables. Therefore, a quadratic optimal control problem is formulated, which is subject to such perturbation differential equations. Thirdly, the Gauss pseudospectral collocation is used to transform the differential and integral operators into algebraic operations. Therefore, an analytical solution of the control correction can be successfully derived in the polynomial space, which comes close to the optimal solution. This method has a fast computation speed and low computational complexity due to the discretization at orthogonal points, making it suitable for online applications. Finally, some simulations and comparisons with the optimal solution and other typical methods have been carried out to evaluate the performance of the method. Results show that it not only performs well in computational efficiency and accuracy but also has great adaptability and optimality. Moreover, Monte Carlo simulations have been conducted. The results demonstrate that it has strong robustness and excellent performance even in highly dispersed environments. Full article

The primary objective is to create a control function that ensures the energetic optimization of a heat pump heating system at any point within the On-Off regulation range. The COPmax value, the optimal performance of the circulation pump, and the actual performance of the compressor are determined based on the water temperature. The objective function is the extended COP equation of the system. The COP equation includes the efficiency of the circulation pump and the compressor. With efficiency considered, the COP is 3.057; without efficiency, it is 3.68. At discrete operating points, steady-state operation is assumed; therefore, the behavior of the components is described using algebraic equations. The equation system was solved in two cycles using numerical iterative Newton linearization and Gaussian elimination methods. First, the mass flow rate was optimized for a water temperature value, then the optimization cycle was repeated at a higher temperature. The temperature increase was 2 °C. Using the values of the optimal performance of the circulation pump and the water temperature, a polynomial control function was developed. Applying the control function, the optimal performance of the circulation pump can be calculated for any operating temperature range. Full article