Lorentzian Form Research Articles

Some Applications of Cauchy-Schwarz Inequality on Lightlike Warped Product Submanifolds

The well-known Cauchy-Schwarz inequality is a fundamental tool of analysis in Hilbert spaces. It is an upper bound on the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is the most important and widely used inequalities in mathematics. We explore that inequality in case of lightlike warped product hypersurface of lorentzian space form to reach to the existence of minimal screen conformal lightlike hypersurface.

Lorentzian Cayley form

Cayley four-form Φ on an eight-dimensional manifold M is a real differential form of a special algebraic type, which determines a Riemannian metric on M as well as a unit real Weyl spinor. It defines a Spin(7) structure on M, and this Spin(7) structure is integrable if and only if Φ is closed. We introduce the notion of a complex Cayley form. This is a one-parameter family of complex four-forms Φτ on M of a special algebraic type. Each Φτ determines a real Riemannian metric on M, as well as a complex unit Weyl spinor ψτ. The subgroup of GL(8,R) that stabilizes Φτ,τ≠0 is SU(4), and Φτ defines on M an SU(4) structure. We show that this SU(4) structure is integrable if and only if Φτ is closed.We carry out a similar construction for the split signature case. There are now two one-parameter families of complex Cayley forms. A complex Cayley form of one type defines an SU(2,2) structure, a form of the other type defines an SL(4,R) structure on M. As in the Riemannian case, these structures are integrable if and only if the corresponding complex Cayley forms are closed. Our central observation is that there exists a special member of the second one-parameter family of complex Cayley forms, which we call the Lorentzian Cayley form. This four-form has the property that it calibrates two four-dimensional subspaces H,H⊥ that have the property that the induced metric on H,H⊥ is Lorentzian. In particular, in a basis adapted to such a calibration, the Lorentzian Cayley form is built from the complex self-dual two-forms for H,H⊥. We explain how these observations solve a certain puzzle that existed in the context of four-dimensional Lorentzian geometry.

Biconservative surfaces with constant mean curvature in lorentzian space forms

Biconservative surfaces with constant mean curvature in lorentzian space forms

Helium atom embedded in non‐ideal classical plasmas: Doubly excited singlet S states

AbstractThe influence of the non‐ideality (NI) of the classical plasmas on the doubly excited singlet S states of the helium atom (He) embedded in the plasma has been investigated theoretically. A pseudopotential containing the Debye length and the non‐ideality parameter (NIP) as its characteristics is used to represent the screened interaction potentials in the plasma. Using a large wavefunction within the framework of the stabilization method, it has been possible to recognize six doubly excited singlet S states (five lying below the He+(2S) excitation threshold and one lying below the He+(3S) excitation threshold) for the plasma‐free case. The energies and the autoionization life‐times of those states are computed by fitting the density of states to the Lorentzian form. Convergence of the computed results is corroborated by increasing the number of terms in the employed wavefunction. For the plasma‐free case, these results are in excellent agreement with the established results in the literature. A comprehensive analysis has been made on changes induced on those doubly excited states by varying NI over a wide range. It has been observed that the energies of the states gradually approach the corresponding threshold energies with the increasing NI of the plasma, whereas the change in the life‐times (alternatively the widths of the states) of the states shows distinctive features depending on the angular momentum of an individual electron.

Spontaneous Emergence of a Causal Time Axis in Euclidean Space from a Gauged Rotational Symmetry Theory

We demonstrate the emergence of an effective “time” axis in the ground state of a gauged rotational symmetry theory in four-dimensional Euclidean space. In so doing, we remove the necessity of Wick rotation to Lorentz spacetime, an arbitrary and sometimes ill-defined procedure, especially for gravity-related theories. We begin by adapting the Cho-Duan-Ge decomposition to the gauge theory of the four-dimensional rotational symmetry group SO(4), where it identifies the maximal Abelian subgroup SO(2)⊗SO(2) in a gauge covariant manner. We then find the one-loop effective theory to have a stable condensate of monopoles corresponding to the reduction of SO(4) symmetry to SO(2)⊗SO(2). The construction of the condensate ensures that the four-dimensional spatial direction of its field strength must coincide with that of this embedding, and that a magnetic potential must be worked against to divert a trajectory away from this direction. Indeed, movement along this direction represents minimal potential energy. We take it to be the time direction. The gauge-dependent nature of the condensate is such that different gauge choices may lead to different time axes and we show on very general grounds that these different coordinate systems must be relatable by transformations of Lorentz form.

Kleinian sphere packings, reflection groups, and arithmeticity

In this paper we study crystallographic sphere packings and Kleinian sphere packings, introduced first by Kontorovich and Nakamura in 2017 and then studied further by Kapovich and Kontorovich in 2021. In particular, we solve the problem of existence of crystallographic sphere packings in certain higher dimensions posed by Kontorovich and Nakamura. In addition, we present a geometric doubling procedure allowing to obtain sphere packings from some Coxeter polyhedra without isolated roots, and study “properly integral” packings (that is, ones which are integral but not superintegral). Our techniques rely extensively on computations with Lorentzian quadratic forms, their orthogonal groups, and associated higher–dimensional hyperbolic polyhedra.

Metrics related to oscillator algebras

A Lie algebra is said to be metric if it admits a symmetric, invariant, and non-degenerate bilinear form. The harmonic oscillator algebra, which arises in the quantum mechanical description of a harmonic oscillator, is the smallest solvable non-abelian metric example. This algebra is the first step of a countable series of solvable Lie algebras that support invariant Lorentzian forms. Generalizing this situation, in this paper, we arrive at the oscillator Lie -algebras as double extensions of metric spaces. The aim of this paper is to present some structural features, invariant metrics, and derivations of this class of algebras and to explore their possibilities of being extended to mixed metric Lie algebras.

Analytical and Numerical Results for the Diffusion-Reaction Equation When the Reaction Coefficient Depends on Simultaneously the Space and Time Coordinates

We utilize the travelling-wave Ansatz to obtain novel analytical solutions to the linear diffusion–reaction equation. The reaction term is a function of time and space simultaneously, firstly in a Lorentzian form and secondly in a cosine travelling-wave form. The new solutions contain the Heun functions in the first case and the Mathieu functions for the second case, and therefore are highly nontrivial. We use these solutions to test some non-conventional explicit and stable numerical methods against the standard explicit and implicit methods, where in the latter case the algebraic equation system is solved by the preconditioned conjugate gradient and the GMRES solvers. After this verification, we also calculate the transient temperature of a 2D surface subjected to the cooling effect of the wind, which is a function of space and time again. We obtain that the explicit stable methods can reach the accuracy of the implicit solvers in orders of magnitude shorter time.

Spatio-temporal non-localities in a solar-like mean-field dynamo

ABSTRACT The scale separation approximation, which is in the base of the solar mean-field dynamo models, can be hardly justified both by observations and theoretical applications to astrophysical dynamos. The general expression for the mean turbulent electromotive force can be written in integral form with convolution of the turbulent effects and mean magnetic field variations over scales of the turbulent flows and global scales of the mean-field dynamo. Following results of direct numerical simulations (DNS), which had been reported earlier, we take the Lorentzian form of the integral convolution kernels as an experimental fact. It allows us to approximate the governing equation for the mean electromotive force by the reaction–diffusion type equation. Solution of the eigenvalue problem reveals a few curious properties of the dynamo model with the non-local mean electromotive force. We find a decrease of the critical dynamo instability threshold, and an increase the dynamo periods of the unstable modes, as reported in earlier studies. Simultaneously, the non-local model shows substantially lower growth rate of the unstable dynamo modes in proximity of the critical threshold than the model which employs the scale separation approximation. We verify these findings using the non-linear solar dynamo model. For the supercritical regime, when the α-effect magnitude is about twice of the instability threshold, the model shows the Parker’s dynamo wave solutions with the wave propagating from the mid-latitude at the bottom of the convection zone towards the solar equator at the surface.

Transformation of 5I8→5F5 Ho absorption band in Ho0.75Nd0.25Fe3(BO3)4 crystal during spin-reorientation transition

Polarized absorption spectra of crystal Ho0.75Nd0.25Fe3(BO3)4 in the region of the 5I8→5F5 absorption band of Ho ions were investigated depending on temperature in the range of 5–20 K. Absorption spectra are decomposed into components of the Lorentz form. The spin-reorientation phase transition was detected at 6.9 K. The observed line position shifts during the reorientation transition were different for different lines, which indicated different local magnetic anisotropy of the Ho ion in its excited states. Significant changes in absorption line intensities as a function of temperature were observed during the reorientation transition and above. These were explained as a consequence of the change in even components of the local crystal field in excited states. These components mix the closely spaced 4f states. Distortion of local symmetry in the ground state was detected at one of the electron transitions.

SPHERICALLY SYMMETRIC SYSTEM OF GRAVITATIONAL AND ELECTROMAGNETIC FIELDS AND THE STRUCTURE OF ITS CONFIGURATION SPACE

Geometrodynamics of charged black holes (BH) described by the system of Maxwell Einstein equations is considered. We start from a spherically symmetric metric, a reduced action, and a Lagrangian written in characteristic variables. The configuration space (CS) metric, Hamiltonian, momentum and electromagnetic constraints are constructed. The system has conservation laws of charge q and mass m. The action functional is transformed into a Jacobi-type functional in CS with a metric conformal to the CS metric. A transformation of field variables is introduced which brings the CS metric to the "Lorentzian"form. The resulting CS metric is the metric of a flat nonholonomic section of a 4-dimensional space. In the new variables, the squared momenta of the system has the Lorentz form. On this basis, quantization is considered. Thanks to the structure of the CS, the momentum operators, the DeWitt equations, and the mass and charge operators are constructed. The equations system of CBH quantum states with certain q and m is constructed. For comparison, we consider the CBH reduced model limited in the T-region. In such the simplified formulation, the T-model equations are integrated and lead to the CBH with continuous spectrum of m and q.

Algebraic equation of motion approach for solving the Anderson model

Based on the algebraic equation of motion (AEOM) approach, we have studied the single-impurity Anderson model by analytically solving the AEOM of the f-electron one-particle Green function in the Kondo limit. The related spectral function satisfies the sum rule and shows that there is a well-known three-peak structure at zero temperature. In the low energy limit, we obtain the analytical formula of the Kondo temperature that is the same as the exact solution in form except for a prefactor. We also show that the shape of the Kondo resonance is the Lorentzian form and the corresponding weight is proportional to the spin-flip correlation function.

Main Curvatures Identities on Lightlike Hypersurfaces of Statistical Manifolds and Their Characterizations

In this study, some identities involving the Riemannian curvature invariants are presented on lightlike hypersurfaces of a statistical manifold in the Lorentzian settings. Several inequalities characterizing lightlike hypersurfaces are obtained. These inequalities are also investigated on lightlike hypersurfaces of Lorentzian statistical space forms.

Waves guided by metasurfaces with non-local impedance boundary conditions

Non-local impedance boundary conditions (IBCs), which consider the spatial dispersion of impedance, simulate material and surface effects in scattering and propagation. These non-local IBCs are often used to describe various metasurfaces. The use of non-local IBCs leads to some problems and contradictions, limiting the areas of their application. In this article, we demonstrate that eigenmodes in a parallel-plate waveguide with non-local IBCs are non-orthogonal. To restore the orthogonality of the waveguide’s eigenmodes, we derive equations for calculating active power flowing along a surface with non-local IBCs. An unambiguous relationship between impedance spatial dispersion and the active power is demonstrated. We also consider the problem of excitation of the parallel-plate waveguide with non-local IBCs. We show that the use of the eigenmode expansion method leads to incorrect results, which can be corrected by an expression of the Lorentz form obtained in this article.

Study of (P, D) Reaction on <sup>209</sup>Bi at 42 Mev

The 209Bi(p,d) 208Bi reaction has been studied with 42 MeV protons. The laboratory angles are 25°, 30°, and 35°. Energy spectra have been calculated theoretically as an incoherent sum of many shell-orbits constituents based on the Distorted-Wave Born Approximation (DWBA). An asymmetric Lorentzian form strength response function having energy-dependent spreading width is adopted in this analysis. The calculated deuteron energy spectrum is reasonably well reproduced in the direct reaction region. The agreement between the experiment and the calculated double differentials cross-sections up to all experimental points is reasonable and satisfactory, thereby yielding confidence in the theoretical models used. J. Bangladesh Acad. Sci. 45(2); 197-203: December 2021

A note on the nonexistence of spacelike hypersurfaces with polynomial volume growth immersed in a Lorentzian space form

We obtain nonexistence results concerning complete noncompact spacelike hypersurfaces with polynomial volume growth immersed in a Lorentzian space form, under the assumption that the support functions with respect to a fixed nonzero vector are linearly related. Our approach is based on a suitable maximum principle recently established by Alías, Caminha and do Nascimento [3].

Localized excitons in ZnMnO

Lines a, b, c, d and impurity absorption edge of ZnMnO for sigma- and π-polarizations of light in temperature interval of 7-300 K were registered in this paper. Intensive lines aπ and asigma are clearly observed in interval of 7-100 K, while other lines are observed only at low temperatures. For determination of type of optical transitions, to which excitonic lines a, b, c, d are corresponding the calculation of oscillator strength of most intensive lines was made. Lines aπ and asigma have Lorentz form, parameters of this form were calculated with OriginPro 9.1 program. The energy of impurity absorption edge was determined. Lines a, b, c and d were analyzed in model of Mn2+-4O2- cluster. Optical transitions take place from antibonding (p+d5)*-states in forbidden gap to state, which is splitted from the bottom of conduction band of ZnMnO. Electron-hole pairs which are localized inside the cluster are called as local excitons. Keywords: zinc oxide, localized excitons, antibonding states.

Локализованные экситоны в ZnMnO

Lines a, b, c, d and impurity absorption edge of ZnMnO for σ-and π-polarizations of light in temperature interval of 7 – 300 К were registered in this paper. Intensive lines aπ and aσ are clearly observed in interval of 7-100 K, while other lines are observed only at low temperatures. For determination of type of optical transitions, to which excitonic lines a, b, c, d are corresponding the calculation of oscillator strength of most intensive lines was made. Lines aπ and aσ have Lorentz form, parameters of this form were calculated with OriginPro 9.1 program. The energy of impurity absorption edge was determined. Lines a, b, c и d were analyzed in model of Mn2+ - 4O2- cluster. Optical transitions take place from antibonding (p+d5)*-states in forbidden gap to state, which is splitted from the bottom of conduction band of ZnMnO. Electron-hole pairs which are localized inside the cluster are called as local excitons.

Complete spacelike hypersurfaces with constant scalar curvature: Descriptions and gaps

We provide sharp lower and upper bounds for the supremum of the norm of the total umbilicity tensor of complete spacelike hypersurfaces with constant scalar curvature immersed in a Lorentzian space form and satisfying a suitable Okumura-type inequality, which corresponds to a weaker hypothesis when compared with the geometric condition of the hypersurface having two distinct principal curvatures. Furthermore, we give a complete description and the gaps of the spacelike hypersurfaces which realize our estimates, obtaining as a consequence new characterizations of totally umbilical spacelike hypersurfaces and hyperbolic cylinders of Lorentzian space forms. Our approach is based on a version of Omori–Yau’s maximum principle for trace-type differential operators defined on a complete Riemannian manifold.

Comparison Geometry for an Extension of Ricci Tensor

For a complete Riemannian manifold M with a (1,1)-elliptic Codazzi self-adjoint tensor field A, we use the divergence type operator \({L_A}(u): = div(A\nabla u)\) and an extension of the Ricci tensor to extend some major comparison theorems in Riemannian geometry. In fact we extend theorems such as mean curvature comparison theorem, Bishop–Gromov volume comparison theorem, Cheeger–Gromoll splitting theorem and some of their famous topological consequences. Also we get an upper bound for the end of manifolds by restrictions on the extended Ricci tensor. The results can be applied to some Riemannian hypersurfaces of Riemannian or Lorentzian space forms.