Search Results (176)

In this study, We explore for Minkowski 3-space ${\mathbb{E}}_1^3$ harmonic surfaces’ geometric features by employing a common tangent vector field along a curve situated on the surface. Our analysis is grounded in the rotation minimizing (RM) Darboux frame, which offers a robust alternative to the classical Frenet frame particularly valuable in the Lorentzian setting, where singularities frequently arise. The RM Darboux frame, tailored to curves lying on surfaces, enables the expression of fundamental invariants such as geodesic curvature, normal curvature, and geodesic torsion. We derive specific conditions that characterize harmonic surfaces based on these invariants. We also clarify the connection between the components of the RM Darboux frame and thesurface’s mean curvature vector. This formulation provides fresh perspectives on the classification and intrinsic structure of harmonic surfaces within Minkowski geometry. To support our findings, we present several illustrative examples that demonstrate the applicability and strength of the RM Darboux approach in Lorentzian differential geometry. Full article

In this article, a high-gain axisymmetric Minkowski fractal reflectarray is designed and fabricated for Ku-Band satellite internet communications. High gain is achieved here by carefully optimising the number of unit cells, their shape modifier, focal length, feed position and scan angle. The space-filling properties of Minkowski fractals help in miniaturising the fractal. The scan angle of the reflectarray varied by adjusting the fractal scaling factor for each unit cell in the array. The reflectarray is symmetric along the X-axis in its design and configuration. Initially, a Minkowski fractal unit cell is designed using iteration-1 in the simulation software. Then, its design parameters are optimised to achieve high gain, a narrow beam, and beam scan capabilities. The sensitivity of design parameters is examined individually using the array synthesis method to achieve these performance parameters. It helps to establish the maximum range of design and performance parameters for this design. The proposed reflectarray resonates at 12 GHz, achieving a gain of over 20 dB and a narrow beamwidth of less than 15 degrees. Finally, the designed fractal reflectarray is tested in real-time simulation environments using MATLAB R2023b, and its performance is evaluated in an interference scenario involving LEO and MEO satellites, as well as a ground station, under various time conditions. For real-world applicability, it is necessary to identify, analyse, and mitigate the unwanted interference signals that degrade the desired satellite signal. The proposed reflectarray, with its performance characteristics and beam scanning capabilities, is found to be an excellent choice for Ku-band satellite internet communications. Full article

In this paper, we introduce and develop the concept of osculating curve pairs in the three-dimensional Minkowski space. By defining a vector lying in the intersection of osculating planes of two non-lightlike curves, we characterize osculating mates based on their Frenet frames. We then derive the transformation matrix between these frames and investigate the curvature and torsion relations under varying causal characterizations of the curves—timelike and spacelike. Furthermore, we determine the conditions under which these generalized osculating pairs reduce to well-known curve pairs such as Bertrand, Mannheim, and Bäcklund pairs. Our results extend existing theories by unifying several known curve pair classifications under a single geometric framework in Lorentzian space. Full article

In this paper, our first objective is to define the idea of grand variable Herz spaces. Then, our main goal is to prove boundedness results for operators, including the rough Riesz potential operator of variable order and the fractional Hardy operators, on grand variable Herz spaces under some proper assumptions. To prove the boundedness results, we use Holder-type and Minkowski inequalities. In the proof of the main result, we use different techniques. We divide the summation into different terms and estimate each term under different conditions. Then, by combining the estimates, we prove that the rough Riesz potential operator of variable order and the fractional Hardy operators are bounded on grand variable Herz spaces. It is easy to show that the rough Riesz potential operator of variable order generalizes the Riesz potential operator and that the fractional Hardy operators are generalized versions of simple Hardy operators. So, our results extend some previous results to the more generalized setting of grand variable Herz spaces. 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, we introduce a new generalized inverse called $\mathfrak{m}$-CCE inverse which presents a generalization of the CCE inverse in Minkowski space by using the $\mathfrak{m}$-core-EP decomposition and the Minkowski inverse. We first show the existence and the uniqueness of the generalized inverse. Then, a number of basic properties and diverse characterizations are derived for the $\mathfrak{m}$-CCE inverse as well as its limit and integral expressions. Additionally, applications of the $\mathfrak{m}$-CCE inverse are given in solving a system of linear equations. Applying the generalized inverse, we introduce a binary relation based on the $\mathfrak{m}$-CCE inverse. Full article

Concrete-like composites are widely used in the building of civil engineering applications such as houses, dams, and roads. Mesoscale modeling is a powerful tool for the physical and mechanical analysis of concrete-like composites. A novel discrete element method (DEM)-enhanced external force-free method for multi-phase concrete-like composite modeling with an interface transition zone (ITZ) is presented in this paper. Firstly, randomly distributed particles with arbitrary shapes are generated based on a grading curve. Then, a Minkowski sum operation for particles is implemented to control the minimum gap between adjacent particles. Secondly, a transition from particles to clumps is realized using the overlapping discrete element cluster (ODEC) method and is randomly placed into a specific space. Thirdly, the DEM simulation with a simple linear contact model is employed to separate the overlapped clumps. Meanwhile, the initial position, displacement, and rotation of clumps are recorded. Finally, the mesoscale model is reconstructed based on the displacement and rotation information. The results show that this method can efficiently generate multi-phase composites with arbitrary particle shapes, high volume fractions, an overlapped ITZ, and a periodic structure. This study proposes a novel, efficient tool for analyzing and designing composite materials in resilient civil infrastructure. Full article

In this paper, we introduce null Cartan normal helices in Minkowski space ${\mathbb{E}}_1^3$. We obtain explicit expressions for their torsions by considering the cases when the C-constant vector field is orthogonal to their axis or not orthogonal to it. We find that the tangent vector field of a null Cartan normal helix satisfies the third-order linear homogeneous differential equation and obtain its general solution in a special case. We prove that null Cartan helices are the only normal helices having two axes and, in a particular case, three axes. Finally, we provide the necessary and sufficient conditions for null Cartan normal helices lying on a timelike surface to be isophotic curves, silhouettes, normal isophotic curves and normal silhouettes with respect to the same axis and provide some examples. Full article

We present a simple argument to derive the transformation of the quantum stochastic calculus formalism between inertial observers and derive the quantum open system dynamics for a system moving in a vacuum (or, more generally, a coherent) quantum field under the usual Markov approximation. We argue, however, that, for uniformly accelerated open systems, the formalism must break down as we move from a Fock representation over the algebra of field observables over all of Minkowski space to the restriction regarding the algebra of observables over a Rindler wedge. This leads to quantum noise having a unitarily inequivalent non-Fock representation: in particular, the latter is a thermal representation at the Unruh temperature. The unitary inequivalence is ultimately a consequence of the underlying flat noise spectrum approximation for the fundamental quantum stochastic processes. We derive the quantum stochastic limit for a uniformly accelerated (two-level) detector and establish an open system description of the relaxation to thermal equilibrium at the Unruh temperature. Full article

We introduced the concept of involute partner-ruled surfaces, which are formed by the involutes of spacelike curves and additional conditions ensuring the presence of definite surface normals in Minkowski three-space. First, we provided the criteria for each couple of involute partner-ruled surfaces to be simultaneously developable and minimal. Then, we established the requirements for the coordinate curves lying on these surfaces to be geodesic, asymptotic, and lines of curvature. We also expanded this paper with an example by providing graphical illustrations of the involute partner-ruled surfaces. Full article

The quantum description of relativistic orientable objects by a scalar field on the Poincaré group is considered. The position of the relativistic orientable object in Minkowski space is completely determined by the position of a body-fixed reference frame with respect to the position of the space-fixed reference frame, so that all the positions can be specified by elements q of the Poincaré group. Relativistic orientable objects are described by scalar wave functions $f\left(q\right)$, where the arguments $q=(x,z)$ consist of space–time points x and of orientation variables z from $SL(2,C)$ matrices. We introduce and study the double-sided representation $\mathit{T}\left(\mathit{g}\right)f\left(q\right)=f\left(g_l^{-1}q{g}_r\right)$, $\mathit{g}=(g_l,g_r)\in \mathit{M},$ of the group $\mathit{M}$. Here, the left multiplication by $g_l^{-1}$ corresponds to a change in a space-fixed reference frame, whereas the right multiplication by $g_r$ corresponds to a change in a body-fixed reference frame. On this basis, we develop a classification of orientable objects and draw attention to the possibility of connecting these results with particle phenomenology. In particular, we demonstrate how one may identify fields described by polynomials in z with known elementary particles of spins 0, ${\displaystyle \frac{1}{2}}$, and 1. The developed classification does not contradict the phenomenology of elementary particles and, in some cases, even provides a group-theoretic explanation for it. Full article

It has been shown that at the present stage of the evolution of the universe, cosmological acceleration is an inevitable kinematical consequence of quantum theory in semiclassical approximation. Quantum theory does not involve such classical concepts as Minkowski or de Sitter spaces. In classical theory, when choosing Minkowski space, a vacuum catastrophe occurs, while when choosing de Sitter space, the value of the cosmological constant can be arbitrary. On the contrary, in quantum theory, there is no uncertainties in view of the following: (1) the de Sitter algebra is the most general ten-dimensional Lie algebra; (2) the Poincare algebra is a special degenerate case of the de Sitter algebra in the limit $R\to \infty$ where R is the contraction parameter for the transition from the de Sitter to the Poincare algebra and R has nothing to do with the radius of de Sitter space; (3) R is fundamental to the same extent as c and ℏ: c is the contraction parameter for the transition from the Poincare to the Galilean algebra and ℏ is the contraction parameter for the transition from quantum to classical theory; (4) as a consequence, the question (why the quantities (c, ℏ, R) have the values which they actually have) does not arise. The solution to the problem of cosmological acceleration follows on from the results of irreducible representations of the de Sitter algebra. This solution is free of uncertainties and does not involve dark energy, quintessence, and other exotic mechanisms, the physical meaning of which is a mystery. Full article

Ruled surfaces in Minkowski 3-space play a crucial role in differential geometry and have significant applications in physics and engineering. This study explores the fundamental properties of ruled surfaces via orthogonal modified frame in Minkowski space ${\mathrm{E}}_1^3$, focusing on their minimality, developability, and curvature characteristics. We examine the necessary and sufficient conditions for a ruled surface to be minimal, considering the mean curvature and its implications. Furthermore, we analyze the developability of such surfaces, determining the conditions under which they can be locally unfolded onto a plane without distortion. The Gaussian and mean curvatures of ruled surfaces in Minkowski space are computed and discussed, providing insights into their geometric behavior. Special attention is given to spacelike, timelike, and lightlike rulings, highlighting their unique characteristics. This research contributes to the broader understanding of the geometric properties of ruled surfaces within the framework of Minkowski geometry. Full article

As shown by Dyson in his famous paper “Missed Opportunities”, it follows, even from purely mathematical considerations, that quantum Poincare symmetry is a special degenerate case of quantum de Sitter symmetries. Thus, the usual explanation of why, in particle physics, Poincare symmetry works with a very high accuracy is as follows. A theory in de Sitter space becomes a theory in Minkowski space when the radius of de Sitter space is very high. However, the answer to this question must be given only in terms of quantum concepts, while de Sitter and Minkowski spaces are purely classical concepts. Quantum Poincare symmetry is a good approximate symmetry if the eigenvalues of the representation operators $M_{4\mu }$ of the anti-de Sitter algebra are much greater than the eigenvalues of the operators $M_{\mu \nu }$ ($\mu ,\nu =0,1,2,3$). We explicitly show that this is the case in the Flato–Fronsdal approach, where elementary particles in standard theory are bound states of two Dirac singletons. Full article

In this paper, we investigate null curves in ${\mathbb{R}}_2^4$, the four-dimensional Minkowski space of index 2, utilizing the concept of hybrid numbers. Hybrid and spatial hybrid-valued functions of a single variable describe a curve in ${\mathbb{R}}_2^4$. We first derive Frenet formulas for a null curve in ${\mathbb{R}}_2^3$, the three-dimensional Minkowski space of index 2, by means of spatial hybrid numbers. Next, we apply the Frenet formulas for the associated null spatial hybrid curve corresponding to a null hybrid curve in order to derive the Frenet formulas for this curve in ${\mathbb{R}}_2^4$. This approach is simpler and more efficient than the classical differential geometry methods and enables us to determine a null curve in ${\mathbb{R}}_2^3$ corresponding to the null curve in ${\mathbb{R}}_2^4$. Additionally, we provide an example of a null hybrid curve, demonstrate the construction of its Frenet frame, and calculate the curvatures of the curve. Finally, we introduce null hybrid Bertrand curves, and by using their symmetry properties, we provide some characterizations of these curves. Full article