442 Results Found

Centroid bodies are a continuous and $GL\left(n\right)$-contravariant valuation and play critical roles in the solution to the Busemann–Petty problem. In this paper, we introduce the notion of harmonic Blaschke–Minkowski homomorphism and show that suc...

We prove that in Minkowski spaces, a harmonic function does not necessarily satisfy the mean value formula. Conversely, we also show that a function satisfying the mean value formula is not necessarily a harmonic function. Finally, we conclude that i...

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 uniqu...

We give a refinement of the converse Hölder inequality for functionals using an interpolation result for Jensen’s inequality. Additionally, we obtain similar improvements of the converse of the Beckenbach inequality. We consider the conver...

The objective of this work is to derive the structure of Minkowski spacetime using a Hermitian spin basis. This Hermitian spin basis is analogous to the Pauli spin basis. The derived Minkowski metric is then employed to obtain the corresponding Loren...

In Lorentz–Minkowski space, the angles between any two non-null vectors have been defined in the sense of the angles in Euclidean space. In this work, the angles relating to lightlike vectors are characterized by the Frenet frame of a pseudo nu...

Schuster introduced the notion of radial Blaschke-Minkowski homomorphism and considered the Busemann-Petty problem for volume forms. Whereafter, Wang, Liu and He presented the $L_p$ radial Blaschke-Minkowski homomorphisms and extended Schuster&...

Homogeneous random fractals form a probabilistic generalisation of self-similar sets with more dependencies than in random recursive constructions. Under the Uniform Strong Open Set Condition we show that the mean D-dimensional (average) Minkowski co...

The existence of solutions to the discrete Orlicz electrostatic q-capacity Minkowski problem was given by Ji and Yang when $1<p<2$. Now, we have studied the problem by removing the assumption that the measure $\mu$ does not have a pair of antipoda...

Equiform geometry is considered an extension of other geometries. Furthermore, an equiform frame is a generalization of the Frenet frame. In this study, we begin by defining the term “equiform parameter (EQP)”, “equiform frame&rdquo...

Based on the fundamental theories of null curves in Minkowski 3-space, the null Darboux mate curves of a null curve are defined which can be regarded as a kind of extension for Bertrand curves and Mannheim curves in Minkowski 3-space. The relationshi...

This work considers the Minkowski sum of an N-dimensional ellipsoid and hyperrectangle, a combination that is extremely relevant due to the usage of ellipsoid-adjacent primitives in computer graphics for work such as 3D Gaussian splatting. While para...

We study the notions of pedal curves, contrapedal curves and $\mathit{B}$-Gauss maps of non-lightlike regular curves in Minkowski 3-space. Then we establish the relationships among the evolutes, the pedal and contrapedal curves. Moreover, we also investigate th...

The causal set program and the Wolfram physics project leave open the problem of how a graph that is a (3+1)-dimensional Minkowski spacetime according to its simple geodesic distances could be generated solely from simple deterministic rules. This pa...

The investigation of objects in Minkowski space is of great significance, especially for those objects with mathematical and physical backgrounds. In this paper, we study nullcone fronts, which are formed by the light rays emitted from points on a sp...

Recently, extensive research has been done on evolute curves in Minkowski space-time. However, the special characteristics of curves demand advanced level observations that are lacking in existing well-known literature. In this study, a special kind...

A list of all possible causal relations in the two-dimensional Minkowski space M is exhausted, based on the duality between timelike and spacelike in this particular case, and thirty topologies are introduced, all of them encapsulating the causal str...

The spherical product of two curves, composed of a total of n components, gives rise to spherical product surfaces in Euclidean space ${\mathbb{E}}^n$, frequently resulting in surfaces of revolution, including superquadrics, which often exhibit inherent symmetry....

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...

By generalizing the notion of the pointwise 1-type Gauss map, the generalized 1-type Gauss map has been recently introduced. Without any assumption, we classified all possible ruled surfaces with the generalized 1-type Gauss map in a 3-dimensional Mi...

A complete perturbation theory suitable for teleparallel gravity is developed. The proposed perturbation scheme takes into account perturbations of the coframe, the metric, and the spin-connection, while ensuring that the resulting perturbed system c...

Standard models of “regular black holes” typically have asymptotically de Sitter regions at their cores. Herein, we shall consider novel “hollow” regular black holes, those with asymptotically Minkowski cores. The reason for d...

We define helical (i.e., helicoidal) hypersurfaces depending on the axis of rotation in Minkowski four-space ${\mathbb{E}}_1^4$ . There are three types of helicoidal hypersurfaces. We derive equations for the curvatures (i.e., Gaussian and mean) and give...

In this paper, we study the optimality conditions for set optimization problems with set criterion. Firstly, we establish a few important properties of the Minkowski difference for sets. Then, we introduce the generalized second-order lower radial ep...

Minkowski Pythagorean hodograph curves are widely studied in computer-aided geometric design, and several methods exist which construct Minkowski Pythagorean hodograph (MPH) curves by interpolating Hermite data in the ${\mathbb{R}}^{2,1}$ Minkowski space. Extending...

We investigate the differences and similarities of the Dirichlet problem of the mean curvature equation in the Euclidean space and in the Lorentz-Minkowski space. Although the solvability of the Dirichlet problem follows standards techniques of ellip...

In the present paper, regular spacelike spatial Minkowski Pythagorean hodograph (MPH) curves are characterized with rational rotation-minimizing frames (RRMFs). We define an Euler–Rodrigues frame (ERF) for MPH curves and by means of this concep...

The current work aims to develop an approximation of the slice of a Minkowski sum of finite number of ellipsoids, sliced up by an arbitrarily oriented plane in Euclidean space ${\mathbb{R}}^3$ that, to the best of the author’s knowledge, has not been address...

This paper presents a spectral calculus for computing the spectra of causal Lorentz invariant Borel complex measures on Minkowski space, thereby enabling one to compute their densities with respect to Lebesque measure. The spectra of certain elementa...

In this work, the embankment surfaces with pseudo null base curves are investigated in Minkowski 3-space. The representation formula of pseudo null curves is obtained via the defined structure functions and the k-type pseudo null helices are discusse...

The problem of classification for imbalanced datasets is frequently encountered in practical applications. The data to be classified in this problem are skewed, i.e., the samples of one class (the minority class) are much less than those of other cla...

In this paper, we analyze the intrinsic geometry of lightlike planes in the three-dimensional Lorentz–Minkowski space ${\mathbb{M}}^3$. We connect the theory of curves lying in lightlike planes in ${\mathbb{M}}^3$ with the theory of curves in the simply isotropic plane ${\mathbb{I}}^2$...

In special relativity, particle trajectories, whether mass-bearing or not, can be traced on the Minkowski spacetime manifold in (3+1)D. Meantime, in quantum mechanics, trajectories in the phase space are not strictly outlined because coordinate and l...

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 th...

Three-dimensional images of multiphase natural and engineered materials obtained by X-ray micro-computed tomography require quantitative processing methods that can describe not only phase volume but also connectivity, spatial heterogeneity, and anis...

We consider Ulisse Dini-type helicoidal hypersurfaces with timelike axis in Minkowski 4-space ${\mathbb{E}}_1^4$ . Calculating the Gaussian and the mean curvatures of the hypersurfaces, we demonstrate some special symmetries for the curvatures when they...

We introduce the generalized helical hypersurface having a space-like axis in five-dimensional Minkowski space. We compute the first and second fundamental form matrices, Gauss map, and shape operator matrix of the hypersurface. Additionally, we comp...

The aim of this paper was to obtain Gauss–Bonnet theorems on the Lorentzian Heisenberg group and the Lorentzian group of rigid motions of the Minkowski plane. At the same time, the sub-Lorentzian limits of Gaussian curvature for surfaces which are $C^2$...

In this paper, the generalized helical hypersurfaces $\mathbf{x}=\mathbf{x}(u,v,w)$ with a time-like axis in Minkowski spacetime ${\mathbb{E}}_1^4$ are considered. The first and the second fundamental form matrices, the Gauss map, and the shape operator matrix of $\mathbf{x}$ are calculated. Mor...

Complex metrics play a fundamental role in applied mathematical analysis and in the study of physical phenomena. In this work, two complex Minkowski spacetime metrics are examined in detail. The investigation centers on the spin matrices that generat...

In this paper, we obtain reverse Minkowski inequalities pertaining to new weighted generalized fractional integral operators. Moreover, we derive several important special cases for suitable choices of functions. In order to demonstrate the efficienc...

Canal surfaces are defined and divided into nine types in Minkowski 3-space ${\mathbb{E}}_1^3$ , which are obtained as the envelope of a family of pseudospheres ${\mathbb{S}}_1^2$ , pseudohyperbolic spheres ${\mathbb{H}}_0^2$ , or lightlike cones ${\mathbb{Q}}^2$ , wh...

In this paper, we consider the pedal curves of the mixed-type curves in the Lorentz–Minkowski plane ${\mathbb{R}}_1^2$. The pedal curve is always given by the pseudo-orthogonal projection of a fixed point on the tangent lines of the base curve. For a mixed-type cur...

Measuring and testing association between categorical variables is one of the long-standing problems in multivariate statistics. In this paper, I define a broad class of association measures for categorical variables based on weighted Minkowski dista...

In differential geometry, relations between curves are a large and important area of study for many researchers. Frame areas are an important tool when studying curves, specially the Frenet–Serret frame along a space curve and the Darboux frame along...

A Regge–Wheeler analysis is performed for a novel black hole mimicker ‘the regular black hole with asymptotically Minkowski core’, followed by an approximation of the permitted quasi-normal modes for propagating waveforms. A first-order WKB approxima...

This article considers a general family of weighted fractional integral operators and utilizes this general operator to establish numerous reverse Minkowski inequalities. When it comes to understanding and investigating convexity and inequality, symm...

In this article, we obtain certain novel reverse Hölder- and Minkowski-type inequalities for modified unified generalized fractional integral operators (FIOs) with extended unified Mittag–Leffler functions (MLFs). The predominant results o...

In this paper, we analyze involutes of pseudo-null curves in Lorentz–Minkowski 3-space. Pseudo-null curves are spacelike curves with null principal normals, and their involutes can be defined analogously as for the Euclidean curves, but they exhibit...

We investigate two physical systems within a spacetime region affected by the nontrivial topology. The set-up for our analysis is a Minkowski metric perturbed by elements reflecting the topological nontriviality. These elements arise when exploring C...