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In that paper, firstly, we get two additional different non‐null space curve evolutions in Minkowski 3‐space by considering Landau–Lifshitz (LL) equation where we identify the spin vector with the binormal vector and the normal vector of these curves, respectively. Then, we obtain some links for constructing the moving non‐null space curves by using the integrable LL equation. Finally, we give as an application the exact solution of the moving non‐null curve evolutions obtained by taking the spin vector, the normal vector of the curve, and we showed graphically that these solutions are wave solutions. [ABSTRACT FROM AUTHOR]

We present the proof for the source of exchange flying ring of the biological bosonic state in homochirality of L-amino acids. It is a source of knot in parallel transport of Yang–Mills field in genetic code evolved from natural selection. It serves as a source of protein folding structure in L-amino acids over all the protein structure of a living organism. In the proof, we modified Frank’s model for homochirality and added more properties of nonlinearity and supersymmetry. The mirror symmetries transform their left and right homochiral hidden states of reversed reaction over Frank’s equation for spontaneous autocatalysis. We also change the rate of reaction in the model from the Euclidean norm to Minkowski metric with Lorentz invariant in special relativity theory. The speed of light in Minkowski space in this model is an analogy with a source of all species of all living organisms with common 20 L-amino acids as their constituents. The result of the proof agrees with the existence of L-amino acids in nature. By long-term of evolution, the number of concentration of left homochirality is more than the concentration of right homochirality in amino acids propositional to the Chern–Simons current. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an invariant by image viewpoint changes by applying an important theorem in conformal geometry stating that every surface of the Minkowski space R 3 , 1 leads to an invariant by conformal transformations. For this, we identify the domain of an image to the disjoint union of horospheres ∐ α H α of R 3 , 1 by means of the powerful tools of the conformal Clifford algebras. We explain that every viewpoint change is given by a planar similarity and a perspective distortion encoded by the latitude angle of the camera. We model the perspective distortion by the point at infinity of the conformal model of the Euclidean plane described by D. Hestenesand we clarify the spinor representations of the similarities of the Euclidean plane. This leads us to represent the viewpoint changes by conformal transformations of ∐ α H α for the Minkowski metric of the ambient space. [ABSTRACT FROM AUTHOR]

Bound state energy eigensolutions of nine different diatomic molecules, specifically, H2$$ {\mathrm{H}}_2 $$, TiH, N2$$ {\mathrm{N}}_2 $$, CO, NO, ScH, CrH, HCl and LiH, placed in a point‐like global monopole, have been calculated by solving the time‐independent Schrödinger equation (SE). The molecules are confined by the Aharonov–Bohm (AB) flux field and the interaction potential among the charged particles is governed by the combined screened modified Kratzer potential (SMKP) plus Hulthén potential model. Three other special cases of the interaction potential have also been discussed here. Asymptotic iteration method has been used for the mathematical calculations. It is difficult to solve the SE exactly for any non‐zero values of the azimuthal quantum number l$$ l $$ due to the presence of the centrifugal barrier term. The well‐known Pekeris approximation technique for the terms 1r2$$ \frac{1}{r^2} $$ and 1r$$ \frac{1}{r} $$ has taken into consideration for the purpose of performing numerical computations connected to an arbitrary l$$ l $$ state. The outcomes of this study are in reasonable agreement with the results of previous research on SMKP, Kratzer‐Fues potential, and modified Kratzer potential in the Minkowski space. It is observed that the energy spectra of the diatomic molecules suffer considerable change due to the effects of the background AB‐flux field and topological defect parameter. [ABSTRACT FROM AUTHOR]

We introduce generalized Darboux frames along a null Cartan curve lying on a timelike surface in Minkowski space 피13 and define null Cartan normal isophotic and normal silhouette curves in terms of the vector field that lies in the normal plane of the curve and belongs to its generalized Darboux frame of the first kind. We investigate null Cartan normal isophotic and normal silhouette curves with constant geodesic curvature kg$$ {k}_g $$ and constant geodesic torsion τg$$ {\tau}_g $$. We obtain the parameter equations of their axes and prove that such curves are the null Cartan helices or the null Cartan cubics. In particular, we show that null Cartan normal isophotic curves with a non‐zero constant curvatures kg$$ {k}_g $$ and τg$$ {\tau}_g $$ have a remarkable property that they are general helices, relatively normal‐slant helices and isophotic curves with respect to the same axis. We prove that null Cartan cubics lying on a timelike surface are normal isophotic curves with a spacelike axis and normal silhouette curves with a lightlike axis. We obtain the relation between Minkowski Pythagorean hodograph cubic curves and null Cartan normal isophotic and normal silhouette curves. Finally, we give numerical examples of null Cartan normal isophotic and normal silhouette curves obtained by integrating the system of two the first order differential equations under the initial conditions. [ABSTRACT FROM AUTHOR]

In the four-dimensional Minkowski space, hypersurfaces classified as right conoids with a time-like axis are introduced and studied. The computation of matrices associated with the fundamental form, the Gauss map, and the shape operator specific to these hypersurfaces is included in our analysis. The intrinsic curvatures of these hypersurfaces are determined to provide a deeper understanding of their geometric properties. Additionally, the conditions required for these hypersurfaces to be minimal are established, and detailed calculations of the Laplace–Beltrami operator are performed. Illustrative examples are provided to enhance our comprehension of these concepts. Finally, the umbilical condition is examined to determine when these hypersurfaces become umbilic, and also the Willmore functional is explored. [ABSTRACT FROM AUTHOR]

In this paper we will look at the well known interlacing problem, but here we consider the result for Hermitian matrices in the Minkowski space, an indefinite inner product space with one negative square. More specific, we consider the n × n matrix A = [ J u − u ⁎ a ] with a ∈ R , J = J ⁎ and u ∈ C n − 1. Then A is H -selfadjoint with respect to the matrix H = I n − 1 ⊕ (− 1). The canonical form for the pair (A , H) plays an important role and the sign characteristic coupled to the pair is also discussed. [ABSTRACT FROM AUTHOR]

The timelike tube theorem asserts that in quantum field theory without gravity, the algebra of observables in an open set U is the same as the corresponding algebra of observables in its "timelike envelope" E (U) , which is an open set that is in general larger. The theorem was originally proved in the 1960's by Borchers and Araki for quantum fields in Minkowski space. Here we sketch the proof of a version of the theorem for quantum fields in a general real analytic spacetime. Details have appeared elsewhere. [ABSTRACT FROM AUTHOR]

In the realm of the four-dimensional Minkowski space L4, the focus is on hypersurfaces classified as right conoids and defined by light-like axes. Matrices associated with the fundamental form, Gauss map, and shape operator, all specifically tailored for these hypersurfaces, are currently undergoing computation. The intrinsic curvatures of these hypersurfaces are determined using the Cayley-Hamilton theorem. The conditions of minimality are addressed by the analysis. The Laplace-Beltrami operator for such hypersurfaces is computed, accompanied by illustrative examples aimed at fostering a more profound understanding of the involved mathematical principles. Additionally, scrutiny is applied to the umbilical condition, and the introduction of the Willmore functional for these hypersurfaces is presented. [ABSTRACT FROM AUTHOR]

We continue our work [16] on the analysis of spatially global stability of self-similar blowup profiles for semilinear wave equations in the radial case. In this paper we study the Yang-Mills equations in (1 + d) -dimensional Minkowski space. For d ≥ 5 , which is the energy supercritical case, we consider an explicitly known equivariant self-similar blowup solution and establish its nonlinear global-in-space asymptotic stability under small equivariant perturbations. The size of the initial data is measured in terms of, in a certain sense, optimal Sobolev norm above scaling. This result complements the existing stability results in odd dimensions, while for even dimensions it is new. [ABSTRACT FROM AUTHOR]

This study focuses on redesigned the Lorentz–Minkowski space, great importance in both physics and geometry, by employing multiplicative metric concepts (angle, norm, distance etc.). The impact of multiplicative arguments on the Lorentz‐Minkowski space has been elucidated, showcasing how multiplicative vectors, subspaces, and planes exhibit multiplicative spacelike, timelike, or lightlike characteristics. Also, we conducted comprehensive comparisons with conventional scenarios, elucidating the vectors and their characteristics within this distinct space. To facilitate visual comprehension, illustrative examples and figures have been provided, offering a clearer understanding of the subject matter. [ABSTRACT FROM AUTHOR]

This work explains an internal contradiction (error) in the current understanding of non-Euclidean Minkowski space. This error happens because the imaginary unit i in the Minkowski space is considered as a number. In order to solve this contradiction, it is explained in this work that the imaginary unit i must be consider as an action sign over a vector because only direction of the vector could be imaginary, since imaginary length is nonsense. The concept on the imaginarity of vector refers to the directions of vector (as "plus" or "minus"), but not to the length value and therefore, all numbers could be considered as freely rotatable vectors. Being considered from this position, both the imaginary and real directions (not lengths) of a vector would be consistent for different observers, because they could consider different real axes. [ABSTRACT FROM AUTHOR]

In the present paper, it has been obtained that the fundamental group of n -dimensional Minkowski space with the time topology contains uncountably many copies of the additive group of integers and is not abelian. The result has been first proved for n = 2. Thereafter, it is extended to n > 2 by proving that loops nonhomotopic in M 2 continue to be nonhomotopic in Mn using embedding of M 2 in Mn as a retract through the projection map. [ABSTRACT FROM AUTHOR]

In this paper, we present a comprehensive investigation into the construction of electromagnetic timelike particles within the frame of Minkowski space, employing the Bishop model. Our study involves deriving the fractional derivatives of key Lorentz forces namely Ω (t) , Ω (v 1) , and Ω (v 2) . Furthermore, we delve into the computation of essential normalizing and recursion operators tailored to magnetic vector fields, drawing from the principles outlined in the Bishop model. Additionally, we extend our analysis to determine the Fermi–Walker (FW) fractional derivatives, which play a crucial role in understanding the behavior and dynamics of the normalizing and recursion operators. Through this thorough exploration, we aim to provide valuable insights into the electromagnetic properties of timelike particles within the context of Minkowski space. [ABSTRACT FROM AUTHOR]

The present study introduces an innovative link between integrable equations and the motion of timelike curves within a three-dimensional Minkowski space. This study aims to establish an anology between the modified generalizations of the Heisenberg spin chain model equation, a complex Korteweg–de Vries equation, and the Ablowitz–Kaup–Newell–Segur hierarchy systems of real type, respectively. This is accomplished through the application of specific functions, which are derived based on the curvatures and torsions of three distinct curves and their corresponding Frenet frames in a 3-dimensional Minkowski space. Making use of this method, the geometric derivation of the integrable equation has been demonstrated with success. [ABSTRACT FROM AUTHOR]

We prove that, in Minkowski space, if a spacelike, (n - 1) -convex hypersurface M with constant σ n - 1 curvature has bounded principal curvatures, then M is convex. Moreover, if M is not strictly convex, after an R n , 1 rigid motion, M splits as a product M n - 1 × R. [ABSTRACT FROM AUTHOR]

In this paper, we examine q-Bernstein-Bézier surfaces in Minkowski space- R13 with q as the shape parameter. These surfaces, a generalization of Bézier surfaces, have applications in mathematics, computer-aided geometric design, and computer graphics for the surface formation and modeling. We analyze the timelike and spacelike cases of q-Bernstein-Bézier surfaces using known boundary control points. The mean curvature and Gaussian curvature of these q-Bernstein-Bézier surfaces are computed by finding the respective fundamental coefficients. We also investigate the shape operator dependency for timelike and spacelike q-Bernstein-Bézier surfaces in Minkowski space- R13 , and provide biquadratic and bicubic q-Bernstein-Bézier surfaces as illustrative examples for different values of the shape controlling parameter q. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to examine the condition for a Finsler space with a generalized α , β -metric to be projectively flat. In addition, we establish that the Finsler space with generalized α , β -metric is a flat-parallel Minkowski space and derive the condition under which the β -change for the aforementioned metric is projective. We also explored the projective nature of β -change for various significant Finsler metrics derived from the generalized α , β -metric. [ABSTRACT FROM AUTHOR]

We define a new gauge independent quasi-local mass and energy, and show its relation to the Brown–York Hamilton–Jacobi analysis. A quasi-local proof of the positivity, based on spacetime harmonic functions, is given for admissible closed spacelike 2-surfaces which enclose an initial data set satisfying the dominant energy condition. Like the Wang-Yau mass, the new definition relies on isometric embeddings into Minkowski space, although our notion of admissibility is different from that of Wang and Yau. Rigidity is also established, in that vanishing energy implies that the 2-surface arises from an embedding into Minkowski space, and conversely the mass vanishes for any such surface. Furthermore, we show convergence to the ADM mass at spatial infinity, and provide the equation associated with optimal isometric embedding. [ABSTRACT FROM AUTHOR]

We study a "p-powered" version K n p ⁢ (F ⁢ (R)) of the well-known Bohr radius problem for the family F ⁢ (R) of holomorphic functions f : R → X satisfying ∥ f ∥ < ∞ , where ∥ ⋅ ∥ is a norm in the function space F ⁢ (R) , R ⊂ ℂ n is a complete Reinhardt domain, and X is a complex Banach space. For all p > 0 , we describe in full detail the asymptotic behavior of K n p ⁢ (F ⁢ (R)) , where F ⁢ (R) is: (a) the Hardy space of X-valued holomorphic functions defined in the open unit polydisk 픻 n ; and (b) the space of bounded X-valued holomorphic or complex-valued pluriharmonic functions defined in the open unit ball B ⁢ (l t n) of the Minkowski space l t n . We give an alternative definition of the optimal cotype for a complex Banach space X in the light of these results. In addition, the best possible versions of two theorems from [C. Bénéteau, A. Dahlner and D. Khavinson, Remarks on the Bohr phenomenon, Comput. Methods Funct. Theory 4 2004, 1, 1–19] and [S. Chen and H. Hamada, Some sharp Schwarz–Pick type estimates and their applications of harmonic and pluriharmonic functions, J. Funct. Anal. 282 2022, 1, Paper No. 109254] have been obtained as specific instances of our results. [ABSTRACT FROM AUTHOR]

There are a few results about the global nonlinear stability of nontrivial large solution to quasilinear wave equations. Time-like extremal surface in Minkowski space is an important model of quasilinear wave equation. There are two folds in this paper. Firstly, we get the existence of traveling wave solution to the time-like extremal hypersurface in |$\mathbb {R}^{1+(n+1)}$|⁠ , which can be considered as the generalized Bernstein theorem. For |$n=2$|⁠ , we are also concerned with global stability of traveling wave solutions with speed of light to time-like extremal hypersurface equation in |$1+(2+1)$| dimensional Minkowski space, which is corresponding with quasilinear wave equation in two space dimensions. [ABSTRACT FROM AUTHOR]

A well-known open problem on the behavior of optimal paths in random graphs in the strong disorder regime, formulated by statistical physicists, and supported by a large amount of numerical evidence over the last decade (Braunstein et al. in Phys Rev Lett 91(16):168701, 2003; Braunstein et al. in Int J Bifurc Chaos 17(07):2215–2255, 2007; Chen et al. in Phys Rev Lett 96(6):068702, 2006; Wu et al. in Phys Rev Lett 96(14):148702, 2006) is as follows: for a large class of random graph models with degree exponent τ ∈ (3 , 4) , distances in the minimal spanning tree (MST) on the giant component in the supercritical regime scale like n (τ - 3) / (τ - 1) . The aim of this paper is to make progress towards a proof of this conjecture. We consider a supercritical inhomogeneous random graph model with degree exponent τ ∈ (3 , 4) that is closely related to Aldous's multiplicative coalescent, and show that the MST constructed by assigning i.i.d. continuous weights to the edges in its giant component, endowed with the tree distance scaled by n - (τ - 3) / (τ - 1) , converges in distribution with respect to the Gromov–Hausdorff topology to a random compact real tree. Further, almost surely, every point in this limiting space either has degree one (leaf), or two, or infinity (hub), both the set of leaves and the set of hubs are dense in this space, and the Minkowski dimension of this space equals (τ - 1) / (τ - 3) . The multiplicative coalescent, in an asymptotic sense, describes the evolution of the component sizes of various near-critical random graph processes. We expect the limiting spaces in this paper to be the candidates for the scaling limit of the MST constructed for a wide array of other heavy-tailed random graph models. [ABSTRACT FROM AUTHOR]

Here we prove a global gauge-invariant radiation estimates for the perturbations of the 3 + 1 dimensional Minkowski spacetime in the presence of Yang–Mills sources. In particular, we obtain a novel gauge invariant estimate for the Yang–Mills fields coupled to gravity in a double null framework in the Causal complement of a compact set of a Cauchy slice. A consequence of our result is the global exterior stability of the Minkowski space under coupled Yang–Mills perturbations. A special structure present both in the null Bianchi equations and the null Yang–Mills equations is utilized crucially to obtain the dispersive estimates necessary to conclude the global existence property. Direct use of Bel-Robinson and Yang–Mills stress-energy tensor to obtain the energy estimates is avoided in favor of weighted integration by parts taking advantage of the manifestly symmetric hyperbolic characteristics of null Bianchi and null Yang–Mills equations. Our result holds for any compact semi-simple gauge group. This is the first stability result of Minkowski space including a non-linear source. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel class of magnetic curves and call it pseudo‐solitonic magnetic curves by considering the connection between two significant types of integrable equations, that is, the nonlinear heat system/the nonlinear Schrödinger equation and moving space curves in the Minkowski 3‐space. Later, we construct a very special class of soliton surfaces and call it pseudo‐solitonic magnetic surfaces by considering the effects of the pseudo‐solitonic magnetic flows in the Minkowski 3‐space. These curves and surfaces are very useful since they not only contain geometric and physical features in their inner forms but also their constructions rely on a simple geometric procedure compared to other models. [ABSTRACT FROM AUTHOR]

The involute of a curve is often called the perpendicular trajectories of the tangent vectors of a unit speed curve. Furthermore, the B-Lift curve is the curve acquired by combining the endpoints of the binormal vectors of a unit speed curve. In this study, we investigate the correspondences between the Frenet vectors of a curve's B-lift curve and its involute. We also give an illustration of a helix that resembles space in Lorentzian 3-space and show how to visualize these curves by deriving the B-Lift curve and its involute. [ABSTRACT FROM AUTHOR]

The document is a response to a paper by Gruber, Block, and Montemayor (GBM) on the topic of time. The author admires GBM's inventive experiments but raises concerns about the philosophical foundations of their ideas. They discuss the use of James Hartle's IGUS model to incorporate subjective experience of time within a 4D Minkowski spacetime framework. The author suggests that thinking in terms of second-order cybernetics and considering a wider range of features could help address the problem GBM is discussing. They also question the assumption that our theories of physics and cosmology can provide a straightforward temporal worldview. The author proposes a dual-aspect monist view, where both subjective and objective time arise from a deeper structure. They believe that the resolution to GBM's problem lies in this deeper domain. [Extracted from the article]

We show by matching two flat spaces one in Minkowski coordinates (empty space) and the other in Minkowski coordinates after a special conformal transformation (also empty space) through a bubble with positive and constant surface tension, that the motion of the bubble is hyperbolic. If the surface tension is very big the initial size of the bubble is as small as we wish, so that we can indeed obtain an infinite universe out of empty spaces. The induced space in the bubble is de Sitter type. [ABSTRACT FROM AUTHOR]

In this article, we defined n-binormal operator, n-quasibinormal operator and skew n-binormal operators to Minkowski space M from Hilbert space. And we stickout some conditions for algebraic properties of these operators in Minkowski spaceM. Also we proved two unitary equivalent binormal operators may belongs to the same class of operator in Minkowski Space. [ABSTRACT FROM AUTHOR]

In this paper, we present a method for numerical computing of some characteristic kinds of non‐null curves lying on a non‐degenerate surface in Minkowski space 피13. Namely, we obtain the system of the first‐order ordinary differential equations that correspond to general helix, relatively normal‐slant helix, and isophote curve and integrate it under chosen initial conditions by applying the ode45 function of MATLAB and Runge‐Kutta method. Depending on the kind of curve, we assume that parametric equation of the surface, an axis vector, value of the real cosine or hyperbolic cosine of the corresponding pseudo angle between axis vector and Darboux frame's vector, normal curvature, and geodesic torsion of the curve are given. Finally, we provide the related examples of numerically computed characteristic curves. [ABSTRACT FROM AUTHOR]

In this study, we demonstrate that when three distinct moving space-like curves with the time-like binormal vectors satisfy the generalizations of the Heisenberg spin chain model, these models are equivalent to the third-order system of the AKNS hierarchy and the complex KdV-type equations. [ABSTRACT FROM AUTHOR]

In this paper, we define three types of 2‐ruled hypersurfaces in the Minkowski 4‐space 피14. We obtain Gaussian and mean curvatures of the 2‐ruled hypersurfaces of type‐1 and type‐2 and some characterizations about its minimality. We also deal with the first Laplace–Beltrami operators of these types of 2‐ruled hypersurfaces in 피14. Moreover, the importance of this paper is the definition of these surfaces by using the octonions in 피14. Thus, this is a new idea and makes the paper original. We give an example of 2‐ruled hypersurface constructed by octonion, and we visualize the projections of the images with MAPLE program. Furthermore, the optical fiber can be defined as a one‐dimensional object embedded in the four‐dimensional Minkowski space 피14. Thus, as a discussion, we investigate the geometric evolution of a linearly polarized light wave along an optical fiber by means of the 2‐ruled hypersurfaces in a four‐dimensional Minkowski space. [ABSTRACT FROM AUTHOR]

In the present paper, we study timelike surfaces with parallel normalized mean curvature vector field in the four-dimensional Minkowski space. We introduce special isotropic parameters on each such surface, which we call canonical parameters, and prove a fundamental existence and uniqueness theorem stating that each timelike surface with parallel normalized mean curvature vector field is determined up to a rigid motion in the Minkowski space by three geometric functions satisfying a system of three partial differential equations. In this way, we minimize the number of functions and the number of partial differential equations determining the surface, thus solving the Lund-Regge problem for this class of surfaces. [ABSTRACT FROM AUTHOR]

We prove global stability for the charged scalar field system on a background spacetime, which is close to 1 + 3 -dimensional Minkowski space and whose outward light cones converge to those for the Schwarzs-child metric at null infinity. The key technique to this proof is the use of a modified null frame, depending only on the mass M of the metric, which captures the asymptotic behavior of the metric at future null infinity. Our results are analogous to results obtained in Minkowski space by Lindblad and Sterbenz in (IMRP Int Math Res Pap 2006:52976, 2006) up to a change in coordinates, and will in the sequel be used to prove the full structure of the Einstein–charge scalar field system in these modified harmonic coordinates. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new type of curve called the k-directional Darboux curve. These curves are generated by vector fields that are constructed using the Darboux frame of a given spacelike curve α lying on a timelike surface in Minkowski 3-space E³1 We give the relationships between these curves and their curvatures. In particular, we show how k-directional Darboux curves can be used to classify some special curves, such as helices and slant helices. [ABSTRACT FROM AUTHOR]

It has long been conjectured that for nonlinear wave equations that satisfy a nonlinear form of the null condition, the low regularity well-posedness theory can be significantly improved compared to the sharp results of Smith-Tataru for the generic case. The aim of this article is to prove the first result in this direction, namely for the time-like minimal surface equation in the Minkowski space-time. Further, our improvement is substantial, namely by 3 / 8 derivatives in two space dimensions and by 1 / 4 derivatives in higher dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we construct electromagnetic conformable timelike particles with Bishop model in Minkowski space. Also, we obtain conformable derivatives of Γ t , Γ m 1 , Γ m 2 Lorentz forces. Then, we compute normalizing and recursion operators of magnetic vector fields according to Bishop model. Finally, we determine F–W conformable derivatives for normalizing and recursional operators. [ABSTRACT FROM AUTHOR]

In this paper we are interested in showing an alternative and simple proof for small energy implies regularity to the Cauchy problem of radially symmetric wave maps from the (1 + 2) -dimensional Minkowski space to an arbitrary smooth Riemannian manifold M ⊂ R n with bounded first derivatives of the second fundamental form. Then, combining the classical works of Struwe (Math Z 242:407–414, 2002; Calc Var Partial Differ Equ 16:431–437, 2003), which gave a simple proof for non-concentration of energy to the corresponding problem, a new proof which we think should it be simple can be provided to the result of global regularity for radially symmetric (1 + 2) -dimensional wave maps, which was first obtained by Christodoulou and Tahvildar-Zadeh (Commun Pure Appl Math 46:1041–1091, 1993). Our method relies on basic energy estimates, based on a new div-curl type lemma developed by Zhou ((1 + 2)-dimensional radially symmetric wave maps revisit) and Wang-Zhou (Global well-posedness for radial extremal hypersurface equation in (1 + 3)-dimensional minkowski space-time in critical sobolev space; On proof of the Wei-Yue Ding's conjecture for Schrödinger map flow). [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new type of curve called the k-directional Darboux curve. These curves are generated by vector fields that are constructed using the Darboux frame of a given spacelike curve α lying on a timelike surface in Minkowski 3-space E³1 We give the relationships between these curves and their curvatures. In particular, we show how k-directional Darboux curves can be used to classify some special curves, such as helices and slant helices. [ABSTRACT FROM AUTHOR]

By bifurcation and topological methods, we study the existence/nonexistence and multiplicity of one-sign or nodal solutions of the following k-th mean curvature problem in Minkowski spacetime r N − k v ′ 1 − v ′ 2 k ′ = λ N C N k r N − 1 H k (r , v) in (0 , R) , | v ′ | < 1 in (0 , R) , v ′ (0) = v (R) = 0. As a previous step, we investigate the spectral structure of its linearized problem at zero. Moreover, we also obtain a priori bounds and the asymptotic behaviors of solutions with respect to λ. [ABSTRACT FROM AUTHOR]

Spatially homogeneous Friedmann–Lemaître–Robertson–Walker (FLRW) solutions constitute an infinite dimensional family of explicit solutions of the Einstein–massless Vlasov system with vanishing cosmological constant. Each member expands toward the future at a decelerated rate. These solutions are shown to be nonlinearly future stable to compactly supported spherically symmetric perturbations, in the case that the spatial topology is that of R 3 . The decay rates of the energy momentum tensor components, with respect to an appropriately normalised double null frame, are compared to those around Minkowski space. When measured with respect to their respective t coordinates, certain components decay faster around Minkowski space, while others decay faster around FLRW. [ABSTRACT FROM AUTHOR]

In 1993, Kahn and Kalai famously constructed a sequence of finite sets in d-dimensional Euclidean spaces that cannot be partitioned into less than parts of smaller diameter. Their method works not only for the Euclidean, but for all -spaces as well. In this short note, we observe that the larger the value of p, the stronger this construction becomes. [ABSTRACT FROM AUTHOR]

In this paper, we studied the quantum dynamics of the TiC, NiC, and CuLi diatomic molecules interacting with the Varshni-Eckart potential (VEP) model in topological defects geometry associated with cosmic string under the influence of magnetic and Aharonov-Bohm (AB) flux fields. We employ the Greene-Aldrich approximation scheme in the centrifugal term and determine the approximate eigenvalue solution using the Nikiforov-Uvarov-Functional Analysis (NUFA) method. The numerical energy spectra for topological defects geometry and external fields with this potential are determined. Subsequently, we applied the eigenvalue solution to various cases of the potential model and obtained their analytical solutions. When compared to Minkowski flat space results, the presence of topological defects and external fields eliminates degeneracy and shifts the energy spectra of the diatomic molecules. [ABSTRACT FROM AUTHOR]

We recently proposed a class of type IIB vacua that yield, at low energies, four–dimensional Minkowski spaces with broken supersymmetry and a constant string coupling. They are compactifications with an internal five-torus bearing a five–form flux Φ and warp factors depending on a single coordinate. The breaking of supersymmetry occurs when the internal space includes a finite interval. A probe-brane analysis revealed a gravitational repulsion and a charge attraction of equal magnitude from the left end of the interval, together with a singularity at the other end. Here we complete the analysis revealing the presence, at one end, of an effective O 3 of negative tension and positive five–form charge. We also determine the values of these quantities, showing that T = − Q = Φ , and characterize the singularity present at the other end of the interval, which hosts an opposite charge. Finally, we discuss various forms of the gravity action in the presence of a boundary and identify a self–adjoint form for its fluctuations. Invited contribution to the special issue of Journal of Physics A: Mathematical and Theoretical on 'Fields, Gravity, Strings and Beyond: In Memory of Stanley Deser' [ABSTRACT FROM AUTHOR]

The specific model studied is in the context of the modified measure formulation the string or branes, where tension appear as an additional dynamical degree of freedom. We then consider the signed reparametrization invariant volume element formulation of dynamical strings and branes and find that the dynamical tension can produce positive tensions or negative tensions, corresponding exactly to strings and branes and antistrings and antibranes respectively. The antistrings are realized when a scalar time that defines the modified measure runs in the opposite direction to the world sheet time. For strings with positive tension, both times run in the same direction. The situation resembles the situation in Relativistic Quantum Mechanics with positive and negative energies, proper time of particles running forward with respect of coordinate time, while for antiparticles proper tome runs opposite of coordinate time. An example, where string antistring pair creation takes place in analogy to the pair creation in an external electric field in QED background field, this time in the presence of a background scalar field that couples to the strings and locally changes the tension, the tension field. When many types of strings probing the same region of space are considered this tension scalar is constrained by the requirement of quantum conformal invariance. For the case of two types of strings probing the same region of space with different dynamically generated tensions, there are two different metrics, associated to the different strings. Each of these metrics have to satisfy vacuum Einstein's equations and the consistency of these two Einstein's equations determine the tension scalar. The universal metric, common to both strings generically does not satisfy Einstein's equation. The two string dependent metrics considered here are flat space in Minkowski space and Minkowski space after a special conformal transformation. The limit, where the two string tensions are the same, is studied, it leads to a well defined solution. If the string tension difference between the two types of strings is very small but finite, the approximately homogeneous and isotropic cosmological solution lasts for a long time, inversely proportional to the string tension difference and then the homogeneity and and isotropy of the cosmological disappears and the solution turns into an expanding brane world, where the strings are confined between two expanding bubbles separated by a very small distance at large times. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the properties of timelike and spacelike shifted-knots Bézier surfaces in Minkowski space- E13. These surfaces are commonly used in mathematical models for surface formation in computer science for computer-aided geometric design and computer graphics, as well as in other fields of mathematics. Our objective is to analyze the characteristics of timelike and spacelike shifted-knots Bézier surfaces in Minkowski space- E13. To achieve this, we compute the fundamental coefficients of shifted-knots Bézier surfaces, including the Gauss-curvature, mean-curvature, and shape-operator of the surface. Furthermore, we present numerical examples of timelike and spacelike bi-quadratic (m = n = 2) and bi-cubic (m = n = 3) shifted-knots Bézier surfaces in Minkowski space- E13 to demonstrate the applicability of the technique in Minkowski space. [ABSTRACT FROM AUTHOR]

For bounded domains of Euclidean spaces with piecewise smooth boundary, the integral of outward unit normal vectors of the boundary is zero. In this paper we consider a similar theorem on Minkowski spaces (Minkowski spaces does not mean finite dimensional Banach spaces but finite dimensional vector spaces with pseudo-inner products). We also consider Gauss' divergence theorem on Minkowski spaces, which implies above. Remark that this theorem implies easily the equation to calculate a kind of centroids of hyperbolic simplices. [ABSTRACT FROM AUTHOR]

The adjoint curve of a Frenet curve r=r(s) is defined as the unit speed curve tangent to the principal normal vector field of r. We show that the adjoint curve of a spacelike curve with timelike binormal is a timelike curve. We obtain some relationships between a Frenet curve and its adjoint in Minkowski 3-space. For a given spacelike curve with timelike binormal, we obtain conditions on surfaces that possess the adjoint curve as a common asymptotic, geodesic or curvature line in Minkowski 3-space. We also give examples confirming our theory. [ABSTRACT FROM AUTHOR]

In the present paper, we obtain the fourth Laplace-Beltrami operator of rotational hypersurfaces about spacelike, timelike, and lightlike axes separately in 4-dimensional Lorentz- Minkowski space and prove theorems about fourth Laplace-Beltrami minimality of them. Also, we construct some examples for these rotational hypersurfaces, obtain their fourth Laplace- Beltrami operators and give their visualizations into 3-spaces. [ABSTRACT FROM AUTHOR]

In this article, we define evolutoids and pedaloids of frontals on timelike surfaces in Minkowski 3-space. The evolutoids of frontals on timelike surfaces are not only the generalization of evolutoids of curves in the Minkowski plane but also the generalization of caustics in Minkowski 3-space. As an application of the singularity theory, we classify the singularities of evolutoids and reveal the relationships between the singularities and geometric invariants of frontals. Furthermore, we find that there exists a close connection between the pedaloids of frontals and the pedal surfaces of evolutoids. Finally, we give some examples to demonstrate the results.