Hyperbolic Angle Research Articles

Curvature properties of spacelike hypersurfaces in a RW spacetime

Curvature invariants of both intrinsic and extrinsic nature play a significant role in elucidating the geometry of a spacetime. In particular, these invariants are useful in detecting event horizon of black holes. Notable examples of spacetimes are provided by the generalized Robertson-Walker (GRW) models. An (m+1)-dimensional GRW spacetime is a Lorentzian warped product I×fN endowed with the warped metric gˆ=−dt2+f2(t)g, having as base an open real interval I equipped with the metric −dt2 and as fiber any Riemannian space (N,g) of dimension m, where f is a smooth positive-valued function on I. In this article, we focus our study on the GRW spacetimes having the fiber a Riemannian space with metric g=gk of constant sectional curvature k and denoted by L1m+1(k,f), i.e. RW spacetimes. Using an approach originally developed in Decu et al. (2008) [46], we obtain in this work the lower bound of the (generalized) normalized Casorati curvatures for a spacelike hypersurface H in the GRW spacetime L1m+1(k,f) in terms of the (normalized) scalar curvature of H, the constant sectional curvature k of the fiber, the normal hyperbolic angle of the hypersurface and the warping function f. We also derive the conditions under which this lower bound is reached. We finally apply the results to some basic cosmological models, namely Lorentz-Minkowski, de Sitter, Einstein-de Sitter, anti de Sitter and steady state spacetimes.

The modification of Mohr-Coulomb criteria based on shape function and determination method of undetermined parameters

Rock strength criteria is a key step to estimate the stability of rock engineering, especially, the effect of intermediate principal stress on rock failure should be considered in formation at great depth. However, the mostly used Mohr-Coulomb criterion don't consider the effect of intermediate principal stress, besides, most rock mechanics laboratories do not have the ability to do true triaxial strength, the strength parameters in poly-axial strength criteria are difficult to be determined. In order to solve these problems, based on the least absolute deviation method, the square of the correlation coefficient, absolute difference and mean difference are adopted to verify the fitting effect of 10 strength criteria on 32 groups of true triaxial strength data. On this basis, the elliptical Lode angle shape function, hyperbolic Lode angle shape function, and the Lode angle shape function based on spatial slip plane are used to establish the modified MC strength criteria, which not only meets the requirements of smoothness and convexness, but also can be used widely in many geomaterials. Besides, strength parameters of the established criteria can be decided by conventional triaxial strength experiment, and the prediction accuracy of the modified MC strength criteria for the 32 groups of true triaxial strength data is better than or close to the lowest fitting error of the existing poly-axial strength criteria.

Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds.

Within the synthetic-geometric framework of Lorentzian (pre-)length spaces developed in Kunzinger and Sämann (Ann. Glob. Anal. Geom. 54 (2018), no. 3, 399-447) we introduce a notion of a hyperbolic angle, an angle between timelike curves and related concepts such as timelike tangent cone and exponential map. This provides valuable technical tools for the further development of the theory and paves the way for the main result of the article, which is the characterization of timelike curvature bounds (defined via triangle comparison) with an angle monotonicity condition. Further, we improve on a geodesic non-branching result for spaces with timelike curvature boundedbelow.

High precision structural mapping using edge filters of potential field and remote sensing data: A case study from Wadi Umm Ghalqa area, South Eastern Desert, Egypt

Recently, detecting the edges and geologic structures (deep and shallow) from potential field sources provides a fundamental tool for interpreting geological structures. Therefore, the high precision computation of the edges of potential sources is becoming an essential requirement for studying subsurface structures. In this study, nine common detectors like tilte angle (TA), Theta-map (TM), Horizontal tilt angle (TDX), hyperbolic tilt angle (HTA), Improved tilt angle (ITA), Tilt angle of the total horizontal gradient (TTHG), Tilt angle of the total gradient (TTG), improved Theta-map (ITM), and improved logistic (IL) were applied. These filters have been tested on theoretical models and real data (aeromagnetic) from the Wadi Umm Ghalqa area, South Eastern Desert of Egypt. The obtained results have shown that the improved logistic (IL) filter could detect edges more successfully than the other eight filters. The results of the IL method were used to generate new accurate structural maps for local and deep components of the investigated area. Moreover, surface structures obtained from remote sensing data were considered to control the interpretation of the mapped subsurface lineaments. Hence, the IL method can be utilized as a sharp tool for the high precision structural imaging of the potential data.

A comparative study on different filters for enhancing potential field source boundaries: synthetic examples and a case study from the Song Hong Trough (Vietnam)

Locating the horizontal boundaries of magnetic and gravity sources provides a fundamental tool for interpreting geological structures. Many filters for detecting the source boundaries have been proposed. These filters are generally constructed based on vertical and horizontal gradients of the magnetic and gravity data. This paper presents a comparative study of various boundary detection filters such as gradient amplitude, tilt derivative, gradient amplitude of tilt angle, theta map, horizontal tilt angle, hyperbolic tilt angle, and tilt angle of gradient amplitude. The applicability of each filter was tested on three synthetic gravity data sets. The results show that the use of the tilt angle of gradient amplitude can display the edges more clearly and precisely. Furthermore, the filters are also performed on a real gravity data set from the Song Hong Trough (Vietnam) where the result computed by the tilt angle of gradient amplitude is also in accord with the geology information.

Rigidity of parabolic and stable spacelike hypersurfaces in a generalized Robertson–Walker spacetime

In this paper, we deal with a parabolic stable spacelike hypersurface immersed in a generalized Robertson–Walker spacetime. We consider the support type function as the product between the warping function and the hyperbolic cosine of the hyperbolic angle of the normal vector field with the timelike parallel direction $$\partial _t$$ . We provide some conditions under which the support type function must be constant or, particularly, when the hypersurface must be a slice, yielding a Calabi–Bernstein type result.

Isoptic Curves of Generalized Conic Sections in the Hyperbolic Plane

We recall the notion of generalized hyperbolic angle between proper and improper straight lines, which is only available in Hungarian and Esperanto. Then we summarize the generalized hyperbolic conic sections. After investigation of the real conic sections and their isoptic curves in the hyperbolic plane $\mathbf{H}^2,$ we consider the problem of isoptic curves of generalized conic sections in the extended hyperbolic plane. This problem is widely investigated in the Euclidean plane $\mathbf{E}^2$ but, in the hyperbolic and elliptic planes, there are few results. Furthermore, we determine and visualize the generalized isoptic curves to all hyperbolic conic sections. For our computations, we use the classical models based on the projective interpretation of hyperbolic geometry. In this way, the isoptic curves can be visualized in the Euclidean screen of a computer.

Structural interpretation of SW part of Denizli, Turkey, based on gravity data analysis

Detection of the linear features is a useful task to understand the tectonism of a region. Gravity anomaly data was used to investigate the tectonic structures of SE part of Denizli, western Anatolia. This region is a tectonically active transition zone between Aegean extensional and Anatolian Taurides compressional provinces. Lineaments are appearance of the instantaneous change in the density of the subsurface rocks in gravity data. To determine the lineaments, derivative-based edge detection methods, horizontal gradient, analytic signal, tilt angle and hyperbolic tilt angle applied gravity anomaly data. In defining the lineaments, 3D sediment-basement model, elevation, geology and seismicity data were used comparatively. The sediment-basement topography was modelled by 3D inversion of gravity data. The maximum sediment thickness is defined 2.4 km in the northern part of Acipayam basin and 2.3 km in the southern part of Denizli basin. The earthquake distribution map, geology and elevation maps were provided insight the linear features of the region. As a result of the study, the new tectonic map is attained. Besides the known structures, many new lineaments and some buried faults were defined in the study area.

Spacelike hypersurfaces in standard static spacetimes

In this work we study spacelike hypersurfaces immersed in spatially open standard static spacetimes with complete spacelike slices. Under appropriate lower bounds on the Ricci curvature of the spacetime in directions tangent to the slices, we prove that every complete CMC hypersurface having either bounded hyperbolic angle or bounded height is maximal. Our conclusions follow from general mean curvature estimates for spacelike hypersurfaces. In case where the spacetime is a Lorentzian product with spatial factor of nonnegative Ricci curvature and sectional curvatures bounded below, we also show that a complete maximal hypersurface not intersecting a spacelike slice is itself a slice. This result is obtained from a gradient estimate for parametric maximal hypersurfaces.

The nature and origin of magnetic anomalies over the Gölcük caldera; Isparta; South-Western Turkey

For the first time; ground magnetic survey was performed which data were inverted by means of total horizontal derivative; horizontal gradient analytic signal; and hyperbolic tilt angle techniques to identify subsurface volcanic structures around Isparta city (South-Western Turkey). Here; Golcuk volcanism took place at the apex of the Isparta Angle at the intersection of the Lycian and Antalya nappes. It initiated between 4.0—4.7 my ago mainly as lava extrusions and ended with phreatoplinian eruptions during Quaternary time. The study area is covered by authochtonous and allochthonous units that are intruded by Pliocene and Quaternary Golcuk volcanics and also overlain by pyroclastic fall and flow deposits. The boundaries were revealed for the buried volcanic structure from the edge detection methods. The geometry of the trachytic dome southwestern of the Golcuk Lake and its downward continuation were studied by 2D modelling with the control of the power spectrum depth results applied to the focused anomaly. The azimuthally-averaged logarithmic power spectra plot indicates that the downward continuation of source depth of the trachytic dome reaches up to 850 m. The forward inversion results indicate that the horizontal size of the model for this trachytic dome is 1250 m beneath the surface while it’s surface extension is only about 400 m.

Aeromagnetic data interpretation of east Qattara Depression, Northwest Desert, Egypt

The study area is located at the east of Qattara Depression at the north of the Western Desert of Egypt. The study area contains Abu Gharadig basin, which is the most petroliferous basin in the Egyptian Western Desert. Only three exploratory wells are presented in the study area, showing a thick sediment section overlying basement rocks. Magnetic data have been frequently used in geophysical exploration. Aeromagnetic data are mainly utilized to estimate the depth to the magnetic basement as well as to delineate the possible structures of the study area. The depth to magnetic basement has been estimated using the analytical solution of exponential equations obtained from the Fourier transformation of magnetic data, assuming multi-prisms. The depths obtained from this technique vary from 0.70 to 2.91 km with an average depth of 2.08 km. Local phase filters have been mainly used as edges detector where the possible occurrences structures can be delineated. Hyperbolic tilt angle, second-order tilt angle, and normalized total horizontal derivative (TDX) provide the best results for delineating the possible structures, showing the possible contacts within the basement of the study area. The edge enhancement filters show that the study area has been affected by different structural trends taking E-W, NE-SW, NNE-SSW, N-S, and ENE-WSW directions.

Rodrigues parameters on dual hyperbolic unit sphere

Rodrigues parameters depend on the tangent of the half rotation angle in Euclidean space but in Dual space, dual Rodrigues parameters contain both rotation angle and distance corresponding the shortest distance between the straight lines in ℝ3. In this paper, we give Cayley's formula for the dual hyperbolic spherical motion and explain 3×3 type L-Dual skew symmetric matrices by using properties of this formula. Then, we obtain Rodrigues pa­rameters of dual Hyperbolic unit sphere and show that Rodrigues parameters contain the hyperbolic rotation angle which is being between timelike lines and distance which is the minimal Lorentzian distance between the timelike lines of .

Reconstruction of an AdS radiation/boson star bulk geometry using light-cone cuts

Light-cone cuts have recently been proposed as a method to reconstruct the conformal metric of a holographic spacetime. We explore how additional information about the bulk geometry gets encoded in the structure of these light-cone cuts. In particular, we study how the hyperbolic angle related to a cusp in the light-cone cut encodes information about the matter content of the spacetime. Combining this information with the mass obtained from an integral at the boundary, we provide an explicit numerical example reconstructing the metric for a 4-dimensional static and spherically symmetric spacetime composed by the superposition of a boson star and a gas of radiation in AdS.

On entire f-maximal graphs in the Lorentzian product Gn×R1

In the Lorentzian product G n × R 1 , we give a comparison theorem between the f -volume of an entire f -maximal graph and the f -volume of the hyperbolic H r + under the condition that the gradient of the function defining the graph is bounded away from 1. This condition comes from an example of non-planar entire f -maximal graph in G n × R 1 and is equivalent to the hyperbolic angle function of the graph being bounded. As a consequence, we obtain a Calabi–Bernstein type theorem for f -maximal graphs in G n × R 1 .

Conformal Hyperbolic Numbers and Two-dimensional Finsler Geometry

An abelian group of two-dimensional conformal hyperbolic numbers is investigated. A characteristic equation of a hyperbolic number is derived using the theory of permanents. A conformal multiplier, which depends on components of a hyperbolic number or its hyperbolic angle, is defined, and generalized hyperbolic functions are considered. A geometric representation of the group is introduced. It is shown that modulus of a hyperbolic number gives a metric for a two-dimensional Finsler geometry with a quadratic form and a conformal multiplier. A one-dimensional parameter is derived by requiring that the form of the metric be invariant. A group of parameters and functions of hyperbolic numbers are also investigated.

Computer Aided Design of Centrifugal Fan Impeller Using CATIA Software

In this paper, the 3D model of the centrifugal fan impeller will be obtained using CAD software (CATIA). The blade will be obtained with 4 design methods: constant blade width and variable blade angle; hyperbolic blade width and variable blade angle; hyperbolic blade width and constant blade angle; linear blade width and variable blade angle. A comparative analysis of the weight of the blade, as well as of the weight of the rotor composed by cascade of blades, front and rear shrouds will be presented. Two materials have been used for all the elements of the centrifugal fan impeller: steel and aluminium. For computational purposes the MATLAB programming language will be used. The centrifugal fan blade will be designed as untwisted geometry with =2 reference curves (one on the rear shroud and the other one on the front shroud) with different numbers of reference points ={5, 10, 20, 30, 40, 50}. After running the MATLAB script file, the coordinates of reference points of the blade will be obtained for all 4 blade design methods. The transfer of the reference points coordinates from MATLAB workspace to CATIA has been performed using the automatic surface generation method, via an EXCEL macro file. This is the recommended method especially for a big number of reference points.

Wigner rotation and Thomas precession: geometric phases and related physical theories

We use a vector parameter description of the Lorentz groups in ℝ2,1 and ℝ3,1 to obtain an exact expression for the Thomas factor as a geometric phase. The effect of phase accumulation in Thomas-Wigner precession phenomena is seen as a manifestation of the hyperbolic solid angle theorem. On the infinitesimal level, our description involves affine connections on the noncompact Hopf fibrations U(1) → SU(1, 1) → Δ and SU(2) → PSL(2,ℂ) → H 3. The associated gauge field is a restriction of the familiar Yang-Mills anti-instanton. We also consider the dual compact case, and we discuss generalizations to arbitrary dimensions and applications in various branches of theoretical physics.

Relativistic kinematics formulation of the polarization effects of Jones-Mueller matrices.

The polarization of a partially coherent, transverse electric, electromagnetic plane wave is commonly represented by a Stokes vector. The similarity between Stokes vectors and four-momentum vectors in special relativity (SR) is studied in depth. The set of Stokes vectors naturally possesses a Euclidean and a Lorentzian geometry. The latter is used to express the polarization-altering properties of Jones-Mueller matrices in a simple and elegant way. In particular, it is shown that the action of a diattenuator on a Stokes vector can be understood in terms of the addition law for velocities from SR. An important simplification in the resulting mathematical expressions further arises if the degree of polarization of a Stokes vector is represented by a hyperbolic polarization angle. This then allows us to demonstrate that the output hyperbolic polarization angle is related to a diattenuator hyperbolic polarization angle and the input hyperbolic polarization angle by the hyperbolic law of cosines holding in a hyperbolic triangle.

A NEW WAY TO DIRICHLET PROBLEMS FOR MINIMAL SURFACE SYSTEMS IN ARBITRARY DIMENSIONS AND CODIMENSIONS

In this paper, by considering a special case of the spacelike mean curvature flow investigated by Li and Salavessa [Math. Z. 269 (2011), 697-719], we obtain a condition for the existence of smooth solutions of the Dirichlet problem for the minimal surface equation in arbitrary codimension. We also show that our condition is sharper than Wang's [Comm. Pure Appl. Math. 57 (2004), 267-281] provided that the hyperbolic angle θ of the initial spacelike submanifold M0 satisfies maxM0 cosh θ> √2.

The Higgs oscillator on the hyperbolic plane and light-front holography

The Light Front Holographic (LFH) wave equation, which is the conformal scalar equation on the plane, is revisited from the perspective of the supersymmetric quantum mechanics, and attention is drawn to the fact that it naturally emerges in the small hyperbolic angle approximation to the "curved" Higgs oscillator on the hyperbolic plane, i.e. on the upper part of the two-dimensional hyperboloid of two sheets, a space of constant negative curvature. Such occurs because the particle dynamics under consideration reduces to the one dimensional Schr\"odinger equation with the second hyperbolic P\"oschl-Teller potential, whose flat-space (small-angle) limit reduces to the conformally invariant inverse square distance plus harmonic oscillator interaction, on which LFH is based. In consequence, energies and wave functions of the LFH spectrum can be approached by the solutions of the Higgs oscillator on the hyperbolic plane in employing its curvature and the potential strength as fitting parameters. Also the proton electric charge form factor is well reproduced within this scheme by means of a Fourier-Helgason hyperbolic wave transform of the charge density. In conclusion, in the small angle approximation, the Higgs oscillator on the hyperbolic plane is demonstrated to satisfactory parallel essential outcomes of the Light Front Holographic QCD. The findings are suggestive of associating the hyperboloid curvature of the with a second scale in LFH, which then could be employed in the definition of a chemical potential.