Radial Part Research Articles

In horizon penetrating coordinates: Kerr black hole metric perturbation, construction and completion

We investigate the Teukolsky equation in horizon-penetrating coordinates to study the behavior of perturbation waves crossing the outer horizon. For this purpose, we use the null ingoing/outgoing Eddington–Finkelstein coordinates. The first derivative of the radial equation is a Fuchsian differential equation with an additional regular singularity to the ones the radial one has. The radial functions satisfy the physical boundary conditions without imposing any regularity conditions. We also observe that the Hertz-Weyl scalar equations preserve their angular and radial signatures in these coordinates. Using the angular equation, we construct the metric perturbation for a circularly orbiting perturber around a black hole in Kerr spacetime in a horizon-penetrating setting. Furthermore, we completed the missing metric pieces due to the mass M and angular momentum J perturbations. We also provide an explicit formula for the metric perturbation as a function of the radial part, its derivative, and the angular part of the solution to the Teukolsky equation. Finally, we discuss the importance of the extra singularity in the radial derivative for the convergence of the metric expansion.

Acceptable solutions of the radial Schrödinger equation for a particle in a central potential

We revisit the discussion about the boundary condition at the origin in the Schrödinger radial equation for central potentials. We give a transparent and convincing reason for demanding the radial part R(r) of the wave function to be finite at r = 0, showing that if R(0) diverges the complete wave function ψ does not satisfy the full Schrödinger equation. If R(r) is singular, we show that the corresponding ψ follows an equation similar to Schrödinger's, but with an additional term involving the Dirac delta function or its derivatives at the origin. Although, in general, understanding some of our arguments requires certain knowledge of the theory of distributions, the important case of a behavior R ∝ 1/r near r = 0, which gives rise to a normalizable ψ, is especially simple: The origin of the Dirac delta term is clearly demonstrated by using a slight modification of the usual spherical coordinates. The argument can be easily followed by undergraduate physics students.

A quantum analog model for a scalar particle interacting with a Kerr–de Sitter black hole

We study the thin-layer quantization of a Schrödinger particle constrained to move along an oblate spheroidal surface embedded in Euclidean space as an analog model to a particle in the background of a black hole and of a wormhole. The Schrödinger equation for the superficial wavefunction leads to an ordinary differential equation with five singularities — four regular and one irregular. We manage to transform this equation into a generalized Heun’s equation, with three regular singular points and one irregular singularity at infinity. This equation was studied by Schäfke and Schmidt back in 1980. By means of an external potential, we are able to reproduce the equations of motion of a particle close both to a black hole and a wormhole, thus creating an effective analog model. Additionally, we present correct and exact solutions for the radial part of the Klein–Gordon equation in the Kerr–de Sitter metric.

Effect of the varying repulsive core of Gay–Berne pair potential on the structure and freezing transition in a two-dimensional system of soft ellipses

ABSTRACT We have investigated the effect of the strength of repulsive core of Gay–Berne (GB) ( n − 6 ) pair potential on the pair correlations and isotropic-nematic ( I − N ) freezing transition in a system of soft ellipses confined to a two-dimensional plane. The power n has been varied in the range 8 ≤ n ≤ 16 at an interval of two. Percus–Yevick (PY) integral equation theory (IET) has been solved to calculate the pair correlations of the isotropic fluid, which has been then used in the density functional theory to determine the isotropic-nematic transition parameters. The pair correlations obtained by PY theory have been compared with those obtained by NVT Monte-Carlo (MC) simulation. We observe a better agreement between PY and MC correlations at lower n -values of the interaction potential. An increase in the values of n results in more intricate structures manifested in both the radial distribution part and the higher order correlations. The I − N transition parameters are found to be sensitive to the value of the exponent n . The transition density is found to decrease, whereas the corresponding orientational order parameter is found to increase with the increase in n for a given temperature. At a given temperature, we observe a cut-off of n , below which the nematic does not stabilise. The smaller value of change in density at the transition indicates that the I − N transition is of weakly first order.

Study of cutting forces during turning of high-hard cast irons

Purpose. Research of cutting force components during machining of high-hard cast iron by special prefabricated cutters with brazed niborite and ciborite inserts. Determination of the influence of geometric characteristics of the cutting tool, cutting modes and hardness of the material being machined on the cutting forces. The methods. The research methods were based on the fundamentals of cutting theory and cutting tools, material science properties of the tool and processed material. A research bench with special equipment was used. Findings. The nature of the change in the components of cutting forces with depth t, feed rate S and cutting speed v was experimentally investigated. The studies have shown an excess of the resulting force in the horizontal plane Pv over the circumferential one Pz, which is the main feature of turning high-hard metals. It has been established that the main limiting factors in the wear of tool cutting inserts are the permissible levels of forces. The hardness of HB has the greatest effect on the radial force Py. Under identical turning conditions for materials with a hardness of 220 to 610 HB, the forces Pz, Py and Px increase by 1,7; 3,3 and 2,3 times, respectively. In addition to the natural increase, the ratio of cutting forces also changes. Reducing the front angle of the blade causes a deterioration in chip formation conditions and leads to an increase in all components of the cutting forces. An increase in the radius at the top of the cutter r causes an increase in the width and decrease in the thickness of the layer to be cut, as well as a decrease in the main angle in the plan φr on the radial part of the blade. The originality. A complex dependence in the structural-stage form was obtained, which gives a clear picture of the influence of each parameter of the turning process on its performance. This dependence is a mathematical model for calculating the optimal operating and geometric parameters of the turning process of high-hard cast irons. The influence of the frontal angle γ, wear value h, and hardness of HB cast iron on the power characteristics during machining of high-hard cast irons has been studied. Practical implementation. It has been shown that it is economically advantageous to maximize the wear of cutting inserts made of relatively expensive composites. In this case, it is necessary to use a technological system of machine-workpiece-tool of high rigidity.

Numerical investigation of the effects of cooling jets on flow and cooling film characteristics downstream of air-cooled integrated flameholder

Integrated flameholders are widely used in advanced augmented/ramjet combustors to reduce the flow loss and weight of aero-engine. Air cooling is a credible method for restraining the increasingly raised wall temperature of flameholders with the increasing inlet temperature and operation cycle of combustors. In this work, an air-cooled integrated flameholder coupled with wall and radial parts was proposed based on our previous studies. Numerical investigations were performed to understand the influence of cooling jets on the flow and cooling film characteristics of air-cooled integrated flameholder in a simplified afterburner. Results suggest that the cooling jet angles α and β on the oblique and rear plates of the wall part mainly affect the flow and cooling film features in the wall backward-facing step, whereas the hole angle θ on the back wall of the radial part has a significant impact on both of the wall step region and downstream of the radial strut. Furthermore, flow analysis of the interaction of radial and wall cooling jets shows that when α and β remain unchanged, θ not only obviously determines the flow features downstream of the radial part but also has a great impact on the flow field in the step region. On the other hand, when θ is constant, α and β almost only change the flow field in the step region, whereas the flow field downstream of the radial part is mainly affected by the flow rate of radial cooling jets. Notably, since the cooling jets on the wall part can remove the flow resistance loss caused by the sudden contraction upon the oblique and rear plates, all air-cooled integrated flameholder schemes proposed in this work can diminish the total pressure loss of afterburner. The total pressure loss is almost unaffected by the varied α and β while θ=90∘ with the flow rate of radial cooling jets increasing. In addition, the total pressure loss is gradually increased with the mainstream velocity increasing and the mainstream temperature decreasing.

Quasinormal Modes of a Charged Black Hole with Scalar Hair

Based on the five-dimensional Einstein–Maxwell theory, Bah et al. constructed a singularity-free topology star/black hole [Phys. Rev. Lett. 126, 151101 (2021)]. After performing the Kaluza–Klein reduction, i.e., integrating the extra space dimension, it can obtain an effective four-dimensional spherically static charged black hole with scalar hair. In this paper, we study the quasinormal modes (QNMs) of the scalar, electromagnetic, and gravitational fields in the background of this effective four-dimensional charged black hole. The radial parts of the perturbed fields all satisfy a Schrödinger-like equation. Using the asymptotic iteration method, we obtain the QNM frequencies semianalytically. For low-overtone QNMs, the results obtained using both the asymptotic iteration method and the Wentzel–Kramers–Brillouin approximation method agree well. In the null coordinates, the evolution of a Gaussian package is also studied. The QNM frequencies obtained by fitting the evolution data also agree well with the results obtained using the asymptotic iteration method.

Polar gravitational waves in f(R, T ϕ ) framework

This work analyzes the propagation of polar gravitational waves for flat FRW cosmic background and in view of f(R, T ϕ ) theory. In what follows, we opt for the Regge-Wheeler polar perturbation scheme in flat geometry of background spacetime and corresponding scalar, field source is also perturbed. The two sets of field equations for flat and perturbed geometry are constructed and furthermore used to evaluate perturbation parameters. The obtained expressions suggest that polar fluctuations affect both geometry and scalar field. The radial and temporal parts of perturbation parameters χ and σ are plotted explicitly while the behavior of ψ is observed via a numerical solution plotted as a 3D graph. The change in these fluctuations for different values of model parameter λ is noticed through these plots.

Radial wave function of 2D and 3D quantum harmonic oscillator

One dimensional quantum harmonic oscillator is well studied in elementary textbooks of quantum mechanics. The wave function of one-dimensional oscillator harmonic can be written in term of Hermite polynomial. Due to the symmetry of the spring energy, the wave functions of two-dimensional and three-dimensional harmonic oscillators can be written as products of the one-dimensional case. Because of that, the wave functions of two- and three-dimensional cases are focused on cartesian coordinates. In this article, we utilize polar and spherical coordinates to describe the wave function of two- and three-dimensional harmonic oscillators, respectively. The radial part of the wave functions can be written in term of associated Laguerre polynomials.

Involving Nikiforov-Uvarov Method in Schrodinger Equation Obtaining Hartmann Potential

The total wave function and the bound state energy are investigated by involving the Nikiforov-Uvarov method to the Schrodinger equation in spherical coordinates employing Hartmann Potential (HP). The HP is considered as the non-central potential that is mostly recognized in nuclear field potentials. Every wave function is specified by a principal quantum number n, angular momentum number l, and magnetic quantum number m. The radial part of the wave function is obtained in terms of the associated Laguerre polynomial, using the coordinate transformation x=cosθ to obtain the angular wave function that depends on inverse associated Legendre polynomials.

Lithium Ion Wave Function (3Li2+) in a Momentum Chamber at n?3

Lithium ion (3Li2+) is a light alkali metal which is reactive and flammable when exposed to water or air. Lithium becomes ionic when lithium metal loses two of its three valence electrons. The wave function is a complex quantity that shows the characteristics of the de Broglie wave. The wave function in question is the radial wave function in the particle, because to find the probability that the electron is in the particle, only the radial part of the wave function is needed. However, there is no literature that discusses the wave function in the momentum space in the case of hydrogenic ions. Therefore, this research will examine the wave function of Lithium ion (3Li2+) in the momentum space. This type of research in the preparation of this article uses non-experimental research. Based on the results of the research that has been done, the wave function of Lithium ion (3 Li2+) in the momentum space at n 3 is obtained. The wave function consists of a radial, angular (polar and azimuth) wave function, as well as the wave function of Lithium ion (3Li2+) as a whole. The fundamental difference between the wave function of Lithium ion (3Li2+) in the momentum space and the wave function of hydrogenic atoms in the other momentum space is the value of the radial function caused by the difference in the value of (z) or atomic number.

Clinical Results of Nonvascularized Metatarsal Articular Bone Grafting for Osteochondral Defects of the Radiocarpal and Distal Radioulnar Joint.

Background Treatment of post-traumatic osteochondral defects in the radiocarpal and distal radioulnar joint with nonvascularized metatarsal articular bone grafting is an option to restore the joint (cartilage) surface and wrist function. Purpose To evaluate the clinical midterm results of 10 consecutive patients who were treated with a nonvascularized metatarsal bone graft for cartilage bone defects of the lunate facet, scaphoid facet, sigmoid notch, or the radial part of the ulnar head. Patients and Methods Patients with isolated osteochondral defects of the lunate facet, scaphoid facet, sigmoid notch, or radial part of the ulnar head, respectively, as a result of wrist trauma were retrospectively identified in a prospectively collected database. The patients symptoms were limited wrist motion and/or pain. Clinical results and complications were extracted from patient's medical files and two questionnaires consisting of the Patient-Rated Wrist Evaluation and additional questions regarding patient satisfaction and return to work and/or hobby. Results Minor complications occurred in three cases and required surgery. In five cases an acceptable donor site foot morbidity was seen at 1 year. One patient was not satisfied due to persistent wrist pain despite adequate range of motion. One patient could not return to its prior work. Another patient could not resume its hobby. Conclusion Retrospective evaluation of 10 cases with resurfacing of the scaphoid facet, sigmoid notch, and/or lunate facet, or radial part of the ulnar head has shown that osseointegration was possible without a vascular pedicle to the graft in all cases with a (mean) follow-up of 5 years. This technique may become an alternative treatment when implants become less available. Level of Evidence Level III.

Investigation of the Strength of a Chain Binder for Securing a Wagon on the Railway Ferry Deck

The article presents the results of the strength analysis of the chain binder components for symmetry fixing a wagon on the railway ferry deck, taking into account the actual hydrometeorological conditions of the navigation area. The simulation of the wagon dynamic load during the railway ferry rolling was carried out to determine the loads acting on the chain binder. The strength of the chain binder hook was analysed. The calculation results showed that the maximum equivalent stresses in the hook occur in the radial part and exceed the allowable ones by 23%. Therefore, to ensure the strength of the chain binder, it is necessary to comply with the appropriate roll angles of the railway ferry or create measures to reduce its load when rolling. The conducted studies will contribute to ensuring the safety of wagon transportation by sea and increasing the efficiency of rail-ferry transportation.

Treatment of Comminuted Radial Head Fractures with Personalized Radial Head Prosthesis Produced with 3D Printing Technology.

The radial head is the forearm's pivot point for supination-pronation motions. Although radial head resection can be performed in comminuted radial head fractures, radial head prosthesis has gained popularity in order to prevent possible complications such as progressive valgus instability in the elbow and secondary ulnocarpal impaction that may develop due to proximal displacement of the radius. As for radial head prostheses, standard prosthetic designs do not provide the same results in everyone, and the alignment of the radial head and radial neck angle cannot be fully achieved. We believe that the radial head, considered the pivot point of the elbow joint, must be reconstructed with a complete anatomical implant. We designed and applied personalized prostheses to our patients, and in this study, we wished to share the 2-year results of these patients. In this study, 23 patients who had comminuted radial head fractures were applied personalized radial head prostheses. Preoperative variables such as type of injury, age, side, additional ligament injury, operation time, number of radial head parts, neck angle of individual prosthesis, stem length, and stem diameter were noted. Mean operating time was 26 ± 9 minutes. One month postoperatively, after physical therapy, elbow ROM, loss of strength compared to the opposite joint, Mayo elbow scores, QuickDASH scores, and patient satisfaction were evaluated. Patients were summoned two years postoperatively. QuickDASH and Mayo elbow scores were satisfactorily good (mean QuickDASH score: 9.091, mean Mayo score: 91.08). According to the results of the patient satisfaction questionnaire, 16 of the 23 patients reported excellent satisfaction, 5 patients good satisfaction, and 2 patients moderate satisfaction. None of the patients reported poor results. Since a personalized radial head prosthesis consists of a single monoblock, its surgical application consists of a single step and has a short operating time. As the applications in the literature increase and more studies are conducted, this subject will be better understood. Our study demonstrated that these patients, whose anatomy were individually replicated, achieved good range of motion and patient satisfaction. In fact, none of the patients reported negative results. In comminuted radial head fractures, surgical implantation of personalized radial head prosthesis is a treatment modality with easy application, short operating time, and good functional results due to nearly complete preservation.

An appearance of classical matter from the self-organizing process of quantum systems

We present a quantum effect where matter follows the classical Hamilton-Jacobi equation, which emerges from quantum systems with Riemannian structures, as in standard quantum systems such as semiconductor heterostructures, quantum plasmas, and quantum dots. The proposed effect is derived from solving a standard elliptic partial differential equation of the radial part of the wave function, which is equivalent to a vanished quantum potential of the system. We then analyze such an effect and examine how the classical matter tends to be denser at the boundary region of the system when the quantum system is given in a finite region in space. While the proposed effect is derived from the hydrodynamical formulation of quantum mechanics, the results are free from any interpretation of quantum mechanics.

An algebraic approach for the Dunkl–Killingbeck problem from the bi-confluent Heun equation

In this paper, we study the Dunkl–Killingbeck problem in two dimensions. We apply the Lie algebraic approach within the framework of quasi-exact solvability to the radial part of the Dunkl–Killingbeck problem to find the general exact expressions for the energies and corresponding wave functions. The allowed values of the potential parameters are the representation space of sl(2) Lie algebra. In addition, we discuss that the effective potential of the Dunkl–Killingbeck is the same as the obtained from the bi-confluent Heun equation by a suitable variable transformation. Following earlier results, we follow the explicit solutions of this differential equation expressed as a series expansion of Hermite functions and obtain the expansion coefficients from a three-term recurrence relation. In the sequel, we present that this construction leads to the known quasi-exactly solvable (QES) form of the Dunkl–Killingbeck problem. Therefore, we find that the expressions for the energy eigenvalues and wave functions of the corresponding potential term are in agreement with those from the QES formalism. Then, we derive the ladder operators for the Dunkl–Killingbeck problem within the algebraic approach. It seems that this method is the Dunkl–Killingbeck rotation problem solved by operators of the su[Formula: see text] Lie algebra in a specific way.

Modeling the Magnetic Field of the Inner Corona in a Radially Expanding Solar Wind

The magnetic field in the interplanetary medium is formed by the action of magnetic field sources on the photosphere of the Sun and currents in the expanding atmosphere of the Sun and the solar wind. In turn, the high-speed plasma flow changes the configuration of the magnetic field lines. The problem of determining the parameters of the magnetic field near the Sun is thus a three-dimensional problem of the interaction of the magnetic field and the plasma of the solar wind. We present analytical expressions for calculating the total magnetic field vector B→(r, θ, ϕ) (in spherical coordinates) for a radially expanding solar wind flow of finite conductivity. The parameters of the solar wind are given in the form of a dimensionless magnetic Reynolds number given as an arbitrary function of the radius, r: Rm = rσμv=ξ(r), where σ, μ, and v denote, respectively, the conductivity, magnetic permeability, and velocity of the solar wind. The solution for the magnetic field components is obtained in the form of a decomposition in spherical functions and a radial part depending on the distance from the Sun. Examples of calculations of the configuration of magnetic fields and structures of the solar corona for the solar eclipse of 21 August 2017 are given.

Neural network method for constructing intermolecular potential energy surfaces of van der Waals complexes

This study proposes a new approach for constructing intermolecular potential energy surfaces (PESs) of van der Waals (vdW) complexes using neural networks. The descriptors utilized in this neural network model are split into two parts: radial parts representing the intermolecular stretching vibrations between monomers and angular parts describing the relative orientation of these molecules. Specifically, the parity-adapted rotational basis functions used in the bound state calculation are taken as the angular descriptors, which ensure the correct symmetry of the PES. The number of orthogonal rotational basis functions is controlled by the maximum value of the angular momentum quantum number. In addition, the symmetry of monomer molecules is achieved by restricting the quantum number of the rotational basis function. The descriptors for five types of van der Waals complexes, including atom-linear, atom-nonlinear, linear-linear, linear-nonlinear and nonlinear-nonlinear molecules complexes, have been derived in this work. The neural network models with these newly developed descriptors were then applied to construct PESs of two van der Waals complexes, Ar-NaCl and N2-OCS. The root-mean-square error values between the fitted and ab initio energies are found to be 0.11 cm−1 and 0.26 cm−1 for Ar-NaCl and N2-OCS, respectively. These results indicate that this method is accurate and effective for constructing high-precision PESs of vdW complexes.

Tactile Perception of the Hand in Children With an Upper Neonatal Brachial Plexus Palsy

ObjectiveTo investigate factors that cause impairment of hand function in children with an upper Neonatal Brachial Plexus Palsy (NBPP), we performed an in-depth analysis of tactile hand sensibility, especially the ability to correctly localize a sensory stimulus on their fingers. DesignA cross-sectional investigation of children with NBPP, compared with healthy controls. The thickest Semmes-Weinstein (SW) monofilament was pressed on the radial or ulnar part of each fingertip (10 regions), while a screen prevented seeing the hand. SettingTertiary referral center for nerve lesions in an academic hospital in The Netherlands. The control group was recruited at their school. ParticipantsForty-one children with NBPP (mean age 10.0 y) and 25 controls (mean age 9.5 y; N=41). InterventionsNot applicable. Main Outcomes MeasuresCorrect localization of the applied stimuli was evaluated, per region, per finger, and per dermatome with a test score. The affected side of the NBPP group was compared with the non-dominant hand of the controls. ResultsThe ability to localize stimuli on the tips of the fingers in children with an upper NBPP was significantly diminished in all fingers, except for the little finger, as compared with healthy controls. Mean localization scores were 6.6 (thumb) and 6.3 (index finger) in the NBPP group and 7.6 in both fingers for controls (maximum score possible is 8.0). Localization scores were significant lower in regions attributed to dermatomes C6 (P<.001) and C7 (P=.001), but not to C8 (P=.115). ConclusionChildren with an upper NBPP showed a diminished and incorrect ability to localize sensory stimuli to their fingers. This finding is likely 1 of the factors underlying the impairment of hand function and should be addressed with sensory focused therapy.

Implementation of genetic and particle swarm optimization algorithm for voltage profile improvement and loss reduction using capacitors in 132 kV Manipur transmission system

Complex power plays a key role in maintaining and sustaining the fields i.e. magnetic and electric fields. The efficiency of the power system depends entirely on the loss and voltage profile level of the system. The lower the level of loss and the higher the voltage profile, the higher the system efficiency. Installation of the capacitors on a system greatly helps in controlling and reducing levels of the reactive energy of the system. The location and size of the capacitors that need to be installed are of great importance in the installation of the bank of capacitors. The idea of a Genetic Algorithm and also a Particle Swarm Algorithm are deployed in this paper for calculating the optimal location and size of the capacitor. In this paper, the system of the 132 KV Manipur Transmission System is designed in ETAP considering a radial distribution part and the optimization method of a genetic method, and also the particle swarm algorithm is utilized for finding the optimal location and size of the capacitor. The system is further optimized by reducing the loss and improving the voltage profile of the system. And the comparison of both the optimization techniques is presented. The results show that PSO provides optimal solutions while GA is simpler in the optimization process.