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We reconstruct the initial functions from the trace of the solution of an initial value problem for the wave equation on the light cone. A method to recover the initial function from the solution of the wave equation on the light cone is already known for odd spatial dimensions. We generalize their work to the fractional wave equation and all dimensions. In other words, we present a method to reconstruct the initial functions from the solution of the fractional wave equation on the light cone. [ABSTRACT FROM AUTHOR]

In this paper, we see that the hypersurfaces L ± in Ando (Abh Math Semin Univ Hambg 92:105–123, 2022, Proposition 1) are neutral but not flat. Nonetheless, we find parallel almost complex structures I ± suitable for Ando (Abh Math Semin Univ Hambg 92:105–123, 2022, Theorem 1) and parallel almost paracomplex structures J ± suitable for Ando (Abh Math Semin Univ Hambg 92:105–123, 2022, Theorem 2). [ABSTRACT FROM AUTHOR]

There has been a great deal of debate as to the role, if any, of Mach's Principle in cosmology. We propose that these questions have meaning only in the context of a cosmic gravitational potential. G4v operates with such a four-potential, which imposes a different symmetry on its treatment of light propagation than that employed by GR. That combination enables a simple cosmic solution that is Lorentz-invariant in free-fall frames of reference. The solution is compared with supernovae data, which, together with an internal self-consistency condition, fixes the parameters of the solution. The resulting Hubble trajectory is, in broad brush, consistent with many cosmic observations. [ABSTRACT FROM AUTHOR]

Statistical surveys show that the majority of traffic accidents occur due to low visibility, highlighting the need to delve into innovative car lighting technologies. A car driver must not only be able to see but also to be seen. The issue of headlight illumination is vital, especially during the dark hours of the night. Therefore, the focus of this article is determining the range of visibility of dipped (low-beam) headlights under specific experimental conditions. We also designed a methodical guideline aimed at identifying the distance at which dipped headlights illuminate the road while a vehicle is in motion. Research conducted on various classes of road confirmed that the Hyundai i40 is best used on higher-class roads, while the Dacia Sandero is better used on lower-class roads due to the shape and spreading out of its light cone. Furthermore, the pros and cons of the distribution of light cones on several classes of road are presented. Sensor-related equipment was also used to investigate light beam afterglow. In particular, an LX-1108 light meter was applied to determine the obstacle illumination intensity, the properties of which enable recording of low lighting values, and a DJI Mavic AIR 2 unmanned aerial vehicle (UAV; drone) was utilized to record the data related to the location of the examined vehicle, as well as light afterglow at night; relevant data evaluation was carried out using Inkscape software. [ABSTRACT FROM AUTHOR]

We consider a novel curvature flow for curves on the 2-dimensional light cone, contained in the 3-dimensional Minkowski space. We show that the solutions to the curvature flow (CF) for such curves are in correspondence with the solutions to the inverse curvature flow (ICF). We prove that the ellipses and the hyperbolas are the only curves that evolve under homotheties. The ellipses are the only closed ones and they are ancient solutions. We show that a spacelike curve on the cone is a self-similar solution to the CF (resp. (ICF)) if, only if, its curvature (resp. inverse of its curvature) differs by a constant c from being the inner product between its tangent vector field and a fixed vector v of the 3-dimensional Minkowski space. The curve is a soliton solution when c = 0 . We prove that for each vector v there exists a 2-parameter family of self-similar solutions to the CF and to the ICF, on the light cone. Considering non-trivial solutions to the CF, we prove that the corresponding solutions to the ICF may have at most three connected components. Moreover, at each end of such a curve the curvature is either unbounded or it tends to 0 or to the constant c. Explicitly given soliton solutions are included and some self-similar solutions on the light cone are visualized. [ABSTRACT FROM AUTHOR]

The intersection of an affine hyperplane in L4 with the light cone C is called a conic section. In this paper, it is proved that the conic sections in L4 are either Riemannian spheres, hyperbolic spaces or horospheres, depending on the causal character of the hyperplane. Analogous results for affine sections of de Sitter and hyperbolic spaces in L4 are also presented at the end. [ABSTRACT FROM AUTHOR]

Statistical surveys show that the majority of traffic accidents occur due to low visibility, highlighting the need to delve into innovative car lighting technologies. A car driver must not only be able to see but also to be seen. The issue of headlight illumination is vital, especially during the dark hours of the night. Therefore, the focus of this article is determining the range of visibility of dipped (low-beam) headlights under specific experimental conditions. We also designed a methodical guideline aimed at identifying the distance at which dipped headlights illuminate the road while a vehicle is in motion. Research conducted on various classes of road confirmed that the Hyundai i40 is best used on higher-class roads, while the Dacia Sandero is better used on lower-class roads due to the shape and spreading out of its light cone. Furthermore, the pros and cons of the distribution of light cones on several classes of road are presented. Sensor-related equipment was also used to investigate light beam afterglow. In particular, an LX-1108 light meter was applied to determine the obstacle illumination intensity, the properties of which enable recording of low lighting values, and a DJI Mavic AIR 2 unmanned aerial vehicle (UAV; drone) was utilized to record the data related to the location of the examined vehicle, as well as light afterglow at night; relevant data evaluation was carried out using Inkscape software.

Heavy mesons and doubly heavy baryons with the dynamic properties governed by the light quark motion relative to the fixed center of forces are studied within the Heavy Quark Effective Theory. The distribution amplitudes of the hadrons in question are compared, model functions of these amplitudes for the doubly heavy baryons are introduced, and their scale dependence is investigated. [ABSTRACT FROM AUTHOR]

This paper discusses the practical development of space-time boundary element methods for the wave equation in three spatial dimensions. The employed trial spaces stem from simplex meshes of the lateral boundary of the space-time cylinder. This approach conforms genuinely to the distinguished structure of the solution operators of the wave equation, so-called retarded potentials. Since the numerical evaluation of the arising integrals is intricate, the bulk of this work is constituted by ideas about quadrature techniques for retarded layer potentials and associated energetic bilinear forms. Finally, we glimpse at algorithmic aspects regarding the efficient implementation of retarded potentials in the space-time setting. The proposed methods are verified by means of numerical experiments, which illustrate their capacity. [ABSTRACT FROM AUTHOR]

For Schrödinger equations with both time-independent and time-dependent Kato potentials, we give a simple proof of the maximal speed bound. The latter states that the probability to find the quantum system outside a ball of radius proportional to the time lapsed decays as an inverse power of time. We give an explicit expression for the constant of proportionality in terms of the maximal energy available to the initial condition. For the time-independent part of the interaction, we require neither decay at infinity nor smoothness. [ABSTRACT FROM AUTHOR]

For the strong gravitational wave model, an explicit transformation is obtained from a privileged coordinate system with a wave variable to a synchronous reference frame with separation of time and space variables. In a synchronous reference frame, a general form of the gravitational wave metric, solutions to the equations of trajectories for test particles in the Hamilton–Jacobi formalism, a solution to the eikonal equation for radiation, and a form of equations for the light "cone" of an observer in a gravitational wave were found. Using the obtained relations, an equation was found for the retarded time of radiation in a gravitational wave. The general relations obtained can be applied both in Einstein's general theory of relativity and in modified theories of gravity. The obtained relations were applied in the work for an exact model of a gravitational wave in the Bianchi type VI universe based on an exact solution of Einstein's vacuum equations. [ABSTRACT FROM AUTHOR]

We analyse the intrinsic and extrinsic geometry of spacelike submanifolds in light cones Λ c (p) of de Sitter and anti-de Sitter spacetimes. Every light cone Λ c (p) contains the lightlike geodesics starting from p and, essentially, it coincides with the horisms E + (p) ∪ E - (p) . The analysis works by means of an explicit correspondence with the spacelike submanifolds through the light cone in the Lorentz–Minkowski spacetime. In particular, a characterization of totally umbilical compact surfaces through light cones in de Sitter and anti-de Sitter is shown and we obtain an estimation of the first eigenvalue of the Laplace operator on a compact spacelike surface in a light cone. [ABSTRACT FROM AUTHOR]