Search Results (123)

This article studies the structure and properties of real-ordered Hilbert spaces, highlighting the roles of the XOR and XNOR logical operators in conjunction with the Yosida and Cayley approximation operators. These fundamental elements are utilized to formulate the Yosida–Cayley Variational Inclusion Problem (YCVIP) and its associated Yosida–Cayley Resolvent Equation Problem (YCREP). To address these problems, we develop and examine several solution methods, with particular attention given to the convergence behavior of the proposed algorithms. We prove both the existence of solutions and the strong convergence of iterative sequences generated under the influence of the aforesaid operators. The theoretical results are supported by a numerical result, demonstrating the practical applicability and efficiency of the suggested approaches. Full article

We present a graphical framework to represent entanglement in three-qubit states. The geometry associated with each entanglement class and type is analyzed, revealing distinct structural features. We explore the connection between this geometric perspective and the tangle, deriving bounds that depend on the entanglement class. Based on these insights, we conjecture a purely geometric expression for both the tangle and Cayley’s hyperdeterminant for non-generic states. As an application, we analyze the energy eigenstates of physical Hamiltonians, identifying the sufficient conditions for genuine tripartite entanglement to be robust under symmetry-breaking perturbations and level repulsion effects. Full article

Very-high-energy electrons, coupled with ultra-high dose rates, are being explored for their potential use in radiotherapy to treat deep-seated tumours. The dose per pulse needed to achieve ultra-high dose rates far exceeds the limit of current medical linear accelerator capabilities. A high dose per pulse has been observed as the limiting factor for many existing dosimeters, resulting in saturation at doses far below what is required. The MOSkin, an existing clinical quality assurance dosimeter, has previously been demonstrated as dose rate independent but has not been subjected to a high dose per pulse. Within this study, the MOSkins dose-per-pulse response was tested for linearity, with a dose per pulse as high as 23 Gy within 200 ns at the ANSTO Australian Synchrotron’s Pulsed Energetic Electrons for Research facility. While using EBT-XD film as a reference dosimeter, a dose rate dependence of the EBT-XD was discovered. Once confirmed and a correction factor established, EBT-XD was used as an independent reference measurement. This work presents confirmation of the MOSkin suitability for ultra-high dose-rate environments with an electron energy of 100 MeV, and a theoretical discussion of its dose-rate and dose-per-pulse independence; the MOSkin is the only detector suitable for both clinical quality assurance, and ultra-high dose-rate measurements in its standard, unmodified form. Full article

An abstract set of one-dimensional (spinor-type) elements randomly oriented on a plane is introduced as a basic subgeometric object. Endowing the set with the binary operations of multiplication and invertible addition sequentially yields a specific semi-group (for which an original Cayley table is given) and a generic algebraic system which is shown to generate, apart from algebras of real and complex numbers, the associative hypercomplex algebras of dual numbers, split-complex numbers, and quaternions. The units of all these algebras turn out to be composed of basic 1D elements, thus ensuring the automatic fulfillment of multiplication rules (once postulated). From the standpoint of a three-dimensional space defined by a vector quaternion triad, the condition of a standard (unit) length of 1D basis elements is considered; it is shown that fulfillment of this condition provides an equation mathematically equivalent to the main equation of quantum mechanics. The similarities and differences of the proposed logical scheme with other approaches that involve abstract subgeometric objects are discussed. Full article

The system of extended ordered XOR-inclusion problems (in short, SEOXORIP) involving generalized Cayley and Yosida operators is introduced and studied in this paper. The solution is obtained in a real ordered Banach space using a fixed-point approach. First, we develop the fixed-point lemma for the solution of SEOXORIP. By using the fixed-point lemma, we develop a three-step iterative scheme for obtaining the approximate solution of SEOXORIP. Under the Lipschitz continuous assumptions of the cost mappings, the strong convergence of the scheme is demonstrated. Lastly, we provide a numerical example with a convergence graph generated using MATLAB 2018a to verify the convergence of the sequence generated by the proposed scheme. Full article

The aim of this work is to develop a discrete-time control algorithm that allows the attitude angle of a satellite with an attached solar panel to track a prescribed periodically changing reference signal with zero asymptotic error. Using the concept of the general regulation theory for the state space setup, combined with a time discretization procedure based on the Cayley–Tustin transformation, we derive an error feedback controller. In our control analysis, we prove and explore several system-theoretic properties that are preserved under this continuous-to-discrete time transformation. The obtained discrete-time controller is then applied as a digital control system, demonstrating zero asymptotic tracking error. The theoretical results are tested on a numerical example and computations are performed within the MATLAB R2024b environment, confirming the highly useful nature of the developed approach. The controller also shows some robustness with respect to parametric uncertainty in the satellite model. Full article

Polyhedral cages (p-cages) provide a good description of the geometry of some families of artificial protein cages. In this paper we identify p-cages made out of two families of equivalent polygonal faces/protein rings, where each face has at least four neighbours and where the holes are contributed by at most four faces. We start the construction from a planar graph made out of two families of equivalent nodes. We construct the dual of the solid corresponding to that graph, and we tile its faces with regular or nearly regular polygons. We define an energy function describing the amount of irregularity of the p-cages, which we then minimise using a simulated annealing algorithm. We analyse over 600,000 possible geometries but restrict ourselves to p-cages made out of faces with deformations not exceeding 10%. We then present graphically some of the most promising geometries for protein nanocages. Full article

Extra edge connectivity is an important parameter for measuring the reliability of interconnection networks. Given a graph G and a non-negative integer h, the h-extra edge connectivity of G, denoted by ${\lambda }_h\left(G\right)$, is the minimum cardinality of a set of edges in G (if it exists) whose deletion disconnects G such that each remaining component contains at least $h+1$ vertices. In this paper, we obtain the h-extra edge connectivity of Cayley graphs generated by transposition trees for $h\le 5$. As byproducts, we derive the h-extra edge connectivity of the star graph $S_n$ and the bubble-sort graph $B_n$ for $h\le 5$. Full article

The spectrum of a graph $\mathrm{\Gamma }$, denoted by $Spec\left(\mathrm{\Gamma }\right)$, is the multiset of eigenvalues of its adjacency matrix. A Cayley graph $Cay(G,S)$ of a finite group G is called Cay-DS (Cayley graph determined by its spectrum) if, for any other Cayley graph $Cay(G,T)$, $Spec\left(Cay\right(G,S\left)\right)=Spec\left(Cay\right(G,T\left)\right)$ implies $Cay(G,S)\cong Cay(G,T)$. A group G is said to be Cay-DS if all Cayley graphs of G are Cay-DS. An interesting open problem in the area of algebraic graph theory involves characterizing finite Cay-DS groups or constructing non-isomorphic Cayley graphs of a non-Cay-DS group that share the same spectrum. The present paper contributes to parts of this problem of metacyclic groups $M_{8p}$ of order $8p$ (with center of order 4), where p is an odd prime, in terms of irreducible characters, which are first presented. Then some new families of pairwise non-isomorphic Cayley graph pairs of $M_{8p}$ ($p\ge 5$) with the same spectrum are found. As a conclusion, this paper concludes that $M_{8p}$ is Cay-DS if and only if $p=3$. Full article

This study proposes a new way to represent elliptic and hyperbolic motions on any general hyperboloids of one or two sheets using the famous Rodrigues, Cayley, and Householder transformations. These transformations are used within the generalized Minkowski 3-space which extends the usual Lorentzian geometry by introducing a generalized scalar product. The study is carried out by considering the unit sphere defined in this generalized space along with the use of three-dimensional generalized Lorentzian skew-symmetric matrices that naturally generate continuous rotational motions. The obtained results provide rotational motions on the sphere in Minkowski 3-space as well as elliptic and hyperbolic motions on general hyperboloids in Euclidean 3-space. A numerical example is provided for each of the explored rotation methods. Full article

In this paper, the stability and Hopf bifurcation of fractional-order quaternion-valued neural networks (FOQVNNs) with various types of time delays are studied. The fractional-order quaternion neural networks with time delays are decomposed into an equivalent complex-valued system through the Cayley–Dickson construction. The existence and uniqueness of the solution for the considered fractional-order delayed quaternion neural networks are proven by using the compression mapping theorem. It is demonstrated that the solutions of the involved fractional delayed quaternion neural networks are bounded by constructing appropriate functions. Some sufficient conditions for the stability and Hopf bifurcation of the considered fractional-order delayed quaternion neural networks are established by utilizing the stability theory of fractional differential equations and basic bifurcation knowledge. To validate the rationality of the theoretical results, corresponding simulation results and bifurcation diagrams are provided. The relationship between the order of appearance of bifurcation phenomena and the order is also studied, revealing that bifurcation phenomena occur later as the order increases. The theoretical results established in this paper are of significant guidance for the design and improvement of neural networks. Full article

This paper is devoted to the analysis of a system of generalized variational inclusion problems involving $\alpha$-averaged and Cayley operators within the framework of a q-uniformly smooth Banach space. We demonstrate that the problem can be reformulated as an equivalent fixed-point equation and propose an iterative method based on the fixed-point approach to obtain the solution. Furthermore, we establish the existence of solutions and analyze the convergence properties of the proposed algorithm under suitable conditions. To validate the effectiveness of the proposed iterative method, we provide a numerical result supported by a computational graph and a convergence plot, illustrating its performance and efficiency. Full article

In this article, we introduce and study a generalized Cayley variational inclusion problem incorporating XOR and XNOR operations. We establish an equivalent fixed-point formulation and demonstrate the Lipschitz continuity of the generalized Cayley approximation operator. Furthermore, we analyze the existence and convergence of the proposed problem using an implicit iterative algorithm. The iterative algorithm and numerical results presented in this study significantly enhance previously known findings in this domain. Finally, a numerical result is provided to support our main result and validate the proposed algorithm using MATLAB programming. Full article

In this paper, we present mathematical geometric models and recursive algorithms to generate and design complex patterns using fractal structures. By applying analytical, iterative methods, iterative function systems (IFS), and L-systems to create geometric models of complicated fractals, we developed fractal construction models, visualization tools, and fractal measurement approaches. We introduced a novel recursive fractal modeling (RFM) method designed to generate intricate fractal patterns with enhanced control over symmetry, scaling, and self-similarity. The RFM method builds upon traditional fractal generation techniques but introduces adaptive recursion and symmetry-preserving transformations to produce fractals with applications in domains such as medical imaging, textile design, and digital art. Our approach differs from existing methods like Barnsley’s IFS and Jacquin’s fractal coding by offering faster convergence, higher precision, and increased flexibility in pattern customization. We used the RFM method to create a mathematical model of fractal objects that allowed for the viewing of polygonal, Koch curves, Cayley trees, Serpin curves, Cantor set, star shapes, circulars, intersecting circles, and tree-shaped fractals. Using the proposed models, the fractal dimensions of these shapes were found, which made it possible to create complex fractal patterns using a wide variety of complicated geometric shapes. Moreover, we created a software tool that automates the visualization of fractal structures. This tool may be used for a variety of applications, including the ornamentation of building items, interior and exterior design, and pattern construction in the textile industry. Full article

On October 16th 1843, the prominent Irish mathematician Sir William Rowan Hamilton, in an inspired act of vandalism, carved his famous $i^2=j^2=k^2=ijk=-1$ on the Brougham Bridge in Dublin, thus starting a major clash of ideas with the potential to change the course of history. Quaternions, as he called his invention, were quite useful in describing Newtonian mechanics, and as it turned out later—also quantum and relativistic phenomena, which were yet to be discovered in the next century. However, the scientific community did not embrace this new approach with enthusiasm: there was a battle to be fought and Hamilton failed to make a compelling case probably because he was standing alone at the time. Although Quaternions were soon to find useful applications in geometry and physics (with the works of Clifford, Cayley, Maxwell, Einstein, Pauli, and Dirac), the battle seemed lost a few decades after Hamilton’s death. But, a century later computer algorithms turned the tides, and nowadays we are witnessing a revived interest in the subject, prompted by technology. Full article