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Journal articles on the topic 'Generalized velocity'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

Stovas, Alexey, and Sergey Fomel. "Generalized velocity approximation." GEOPHYSICS 84, no. 1 (2019): C27—C40. http://dx.doi.org/10.1190/geo2018-0401.1.

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We have developed a new approximation for P-wave velocity that has the same functional form for phase and group velocities. We call it the generalized velocity approximation (GVA) because it is similar to the generalized moveout approximation. The 2D GVA has five parameters, and the 3D GVA has 12 parameters. Our approximation is exact for the phase velocity in a transversely isotropic medium. The parameters of the 3D version of the proposed approximation are defined in all of the symmetry planes in the same fashion. Numerical examples indicate that our approximation has the same accuracy as re

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2

GÖRSCH, D. "GENERALIZED DISCRETE VELOCITY MODELS." Mathematical Models and Methods in Applied Sciences 12, no. 01 (2002): 49–75. http://dx.doi.org/10.1142/s0218202502001544.

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Starting from a mesoscopic principle of moment conservation, discrete Boltzmann collision operators Jh are constructed, which both converge to bounded collision operators JΩ and have the same collision invariants as the original Boltzmann collision operator J. The crucial point of this construction is the application of a weak formulation of the gain operator to remove the post-collision velocities from it as well as the development of moment conserving integration formulas for the approximation of surface integrals over the unit sphere. Finally two applications for the discrete operators are

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3

Anim-Appiah, K. D. "On generalized covariance-based velocity estimation." IEEE Transactions on Vehicular Technology 48, no. 5 (1999): 1546–57. http://dx.doi.org/10.1109/25.790529.

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4

Swamee, Prabhata K. "Generalized Inner Region Velocity Distribution Equation." Journal of Hydraulic Engineering 119, no. 5 (1993): 651–56. http://dx.doi.org/10.1061/(asce)0733-9429(1993)119:5(651).

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5

ZHU, ChangSheng, HaiJun ZHANG, Qin YANG, and ZhiXian ZHONG. "Generalized maxwell velocity slip boundary model." SCIENTIA SINICA Physica, Mechanica & Astronomica 43, no. 5 (2013): 662–69. http://dx.doi.org/10.1360/132011-827.

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6

Chen, Jing-Bo. "On the selection of reference velocities for split-step Fourier and generalized-screen migration methods." GEOPHYSICS 75, no. 6 (2010): S249—S257. http://dx.doi.org/10.1190/1.3517745.

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Split-step Fourier and generalized-screen migration methods are approximations with separation of variables to the one-way wave operator, in which the separation of space and wavenumber variables makes it possible to use the discrete fast Fourier transform to achieve computational efficiency. Both methods require the selection of reference velocities. Different choices of reference velocities lead to different velocity perturbations, and the smaller the velocity perturbations, the better the final image quality. The benefits of selecting a more representative reference velocity for the split-s

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7

HEPPE, BURKHARD M. O. "Generalized Langevin equation for relative turbulent dispersion." Journal of Fluid Mechanics 357 (February 25, 1998): 167–98. http://dx.doi.org/10.1017/s0022112097008069.

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The relative velocity of two fluid particles in homogeneous and stationary turbulence is considered. Looking for reduced dynamics of turbulent dispersion, we apply the nonlinear Mori–Zwanzig projector method to the Navier–Stokes equations. The projector method decomposes the Lagrangian acceleration into a conditionally averaged part and a random force. The result is an exact generalized Langevin equation for the Lagrangian velocity differences accounting for the exact equation of the Eulerian probability density. From the generalized Langevin equation, we obtain a stochastic model of relative

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8

SAWADA, SHIRO. "GENERALIZED OPTIMAL VELOCITY MODEL FOR TRAFFIC FLOW." International Journal of Modern Physics C 13, no. 01 (2002): 1–12. http://dx.doi.org/10.1142/s0129183102002894.

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A generalized optimal velocity model is analyzed, where the optimal velocity function depends not only on the headway of each car but also the headway of the immediately preceding one. The stability condition of the model is derived by considering a small perturbation around the homogeneous flow solution. The effect of the generalized optimal velocity function is also confirmed with numerical simulations, by examining the hysteresis loop in the headway-velocity phase space, and the relation between the flow and density of cars. In the model with a specific parameter choice, it is found that an

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Grosan, Teodor, Cornelia Revnic, and Ioan Pop. "Blasius Problem with Generalized Surface Slip Velocity." Journal of Applied Fluid Mechanics 9, no. 6 (2016): 1641–44. http://dx.doi.org/10.18869/acadpub.jafm.68.235.24826.

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10

Zhang, Xuelong. "Generalized transform of the relativistic phase velocity." European Journal of Physics 15, no. 6 (1994): 309–11. http://dx.doi.org/10.1088/0143-0807/15/6/005.

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11

Zhang, Lina, Xiaoting Rui, Jianshu Zhang, Junjie Gu, and Xizhe Zhang. "Exploring Energy in the Direct Correction Method for Correcting Geometric Constraint Violations." Mathematics 11, no. 6 (2023): 1510. http://dx.doi.org/10.3390/math11061510.

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The direct correction method is widely used for eliminating geometric constraint violations. This method involves iteratively adjusting the generalized coordinates, which are assumed to be consistent and remain so during the velocity-level corrections. However, the corrected generalized coordinates cause a significant effect on the velocity constraint violations. In this paper, simultaneously correcting both the generalized coordinates and velocities is proposed. A semi-analytic approach to solve the Jacobian matrix, which is used to correct the generalized coordinates and velocities, was empl

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12

PAN, LIUBIN, and PAOLO PADOAN. "Relative velocity of inertial particles in turbulent flows." Journal of Fluid Mechanics 661 (July 27, 2010): 73–107. http://dx.doi.org/10.1017/s0022112010002855.

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We present a model for the relative velocity of inertial particles in turbulent flows that provides new physical insight into this problem. Our general formulation shows that the relative velocity has contributions from two terms, referred to as the ‘generalized acceleration’ and ‘generalized shear’, because they reduce to the well-known acceleration and shear terms in the Saffman–Turner limit. The generalized shear term represents particles' memory of the flow velocity difference along their trajectories and depends on the inertial particle pair dispersion backward in time. The importance of

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13

Carcione, José M., and Fabio Cavallini. "The generalized SH‐wave equation." GEOPHYSICS 60, no. 2 (1995): 549–55. http://dx.doi.org/10.1190/1.1443792.

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We present a generalization of the SH‐wave equation for anisotropic and dissipative media. The most general case in which SH‐waves are decoupled from P‐ and SV‐waves at all propagation angles is that of propagation in the plane of symmetry of a monoclinic medium. In the isotropic case, the SH constitutive equation involves only one elastic coefficient (the rigidity); here, three elastic coefficients are needed. Moreover, dissipation is introduced by using Boltzmann’s law based on several relaxation mechanisms. Anisotropic attenuation and velocity dispersion are guaranteed by choosing different

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14

Fomel, Sergey, and Alexey Stovas. "Generalized nonhyperbolic moveout approximation." GEOPHYSICS 75, no. 2 (2010): U9—U18. http://dx.doi.org/10.1190/1.3334323.

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Reflection moveout approximations are commonly used for velocity analysis, stacking, and time migration. A novel functional form for approximating the moveout of reflection traveltimes at large offsets is introduced. In comparison with the classic hyperbolic approximation, which uses only two parameters (zero-offset time and moveout velocity), this form involves five parameters that can be determined, in a known medium, from zero-offset computations and from tracing one nonzero-offset ray. It is called a generalized approximation because it reduces to some known three-parameter forms with a pa

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Askari, Roohollah, and Robert J. Ferguson. "Dispersion and the dissipative characteristics of surface waves in the generalized S-transform domain." GEOPHYSICS 77, no. 1 (2012): V11—V20. http://dx.doi.org/10.1190/geo2010-0330.1.

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Wavenumber, group velocity, phase velocity, and frequency-dependent attenuation characterize the propagation of surface waves in dispersive, attenuating media. We use a mathematical model based on the generalized [Formula: see text] transform to simultaneously estimate these characteristic parameters for later use in joint inversion for near-surface shear wave velocity. We use a scaling factor in the generalized S transform to enable the application of the method in a highly dispersive medium. We introduce a cost function in the [Formula: see text]-domain to estimate an optimum value for the s

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16

Sutton, G. R., and B. J. Moore. "Velocity modelling using a generalized linear inversion technique." Exploration Geophysics 16, no. 2-3 (1985): 287–88. http://dx.doi.org/10.1071/eg985287.

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17

Crawford, Paulo, and Ismael Tereno. "Generalized Observers and Velocity Measurements in General Relativity." General Relativity and Gravitation 34, no. 12 (2002): 2075–88. http://dx.doi.org/10.1023/a:1021131401034.

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18

Xu, Qiancheng, Mohcine Chraibi, Antoine Tordeux, and Jun Zhang. "Generalized collision-free velocity model for pedestrian dynamics." Physica A: Statistical Mechanics and its Applications 535 (December 2019): 122521. http://dx.doi.org/10.1016/j.physa.2019.122521.

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19

Essex, Christopher, and Indrani Das. "Radiative Transfer and Generalized Wind." Entropy 22, no. 10 (2020): 1153. http://dx.doi.org/10.3390/e22101153.

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Dissimilar flows can be compared by exploiting the fact that all flux densities divided by their conjugate volume densities form velocity fields, which have been described as generalized winds. These winds are an extension of the classical notion of wind in fluids which puts these distinct processes on a common footing, leading to thermodynamical implications. This paper extends this notion from fluids to radiative transfer in the context of a classical two-stream atmosphere, leading to such velocities for radiative energy and entropy. These are shown in this paper to exhibit properties for ra

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20

Abdi, Hamid, Saeid Nahavandi, Yakov Frayman, and Anthony A. Maciejewski. "Optimal mapping of joint faults into healthy joint velocity space for fault-tolerant redundant manipulators." Robotica 30, no. 4 (2011): 635–48. http://dx.doi.org/10.1017/s0263574711000671.

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SUMMARYSelf-reconfiguration of robotic manipulators under joint failure can be achieved via fault-tolerance strategies. Fault-tolerant manipulators are required to continue their end-effector motion with a minimum velocity jump, when failures occur to their joints. Optimal fault tolerance of the manipulators requires a framework that can map the velocity jump of the end-effector to the compensating joint velocity commands. The main objective of the present paper is to propose a general framework for the fault tolerance of the manipulators, which can minimize the end-effector velocity jump. In

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21

Fan, Xuelong, Carl Mikael Lind, Ida-Märta Rhen, and Mikael Forsman. "Effects of Sensor Types and Angular Velocity Computational Methods in Field Measurements of Occupational Upper Arm and Trunk Postures and Movements." Sensors 21, no. 16 (2021): 5527. http://dx.doi.org/10.3390/s21165527.

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Accelerometer-based inclinometers have dominated kinematic measurements in previous field studies, while the use of inertial measurement units that additionally include gyroscopes is rapidly increasing. Recent laboratory studies suggest that these two sensor types and the two commonly used angular velocity computational methods may produce substantially different results. The aim of this study was, therefore, to evaluate the effects of sensor types and angular velocity computational methods on the measures of work postures and movements in a real occupational setting. Half-workday recordings o

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22

Palmer, Derecke. "The measurement of weak anisotropy with the generalized reciprocal method." GEOPHYSICS 65, no. 5 (2000): 1583–91. http://dx.doi.org/10.1190/1.1444846.

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Anisotropy parameters can be determined from seismic refraction data using the generalized reciprocal method (GRM) for a layer in which the velocity can be described with the Crampin approximation for transverse isotropy. The parameters are the standard anisotropy factor, which is the horizontal velocity divided by the vertical velocity, and a second poorly determined parameter which, for weak anisotropy, is approximated by a linear relationship with the anisotropy factor. Although only one anisotropy parameter is effectively determined, the second parameter is essential to ensure that the ani

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23

Stork, Christof, and Robert W. Clayton. "Using constraints to address the instabilities of automated prestack velocity analysis." GEOPHYSICS 57, no. 3 (1992): 404–19. http://dx.doi.org/10.1190/1.1443255.

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Generalized prestack velocity analysis methods that use an automated approach to resolve laterally variable interval velocity fields are beset by a series of problems. The problem of resolving lateral velocity variations has inherent complications that prevent automated methods from being robust enough to be applied routinely to data from a variety of geologic provinces. The use of automated prestack velocity analysis methods will not eliminate the step of carefully producing an initial velocity model derived from regional geologic information and an interpretation of a conventionally processe

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24

SAVCI, Ümit Ziya. "Kinematic Analysis in 3-Dimensional Generalized Space." Cumhuriyet Science Journal 43, no. 2 (2022): 299–307. http://dx.doi.org/10.17776/csj.1054869.

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In this paper, we have first obtained the derivatives of spherical and spatial motions by using the special matrix group in generalized space E3(α,β). The rotation matrices and tangent operators were found by using derivatives of one- and multi-parameters motions in E3(α,β). Also, we obtained the angular velocity matrix of the moving body and its linear velocity vector. Finally, we gave some examples including applications of tangent operators and rotation matrices in support of our results.

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25

Peidro, Adrian, and Edward J. Haug. "Obstacle Avoidance in Operational Configuration Space Kinematic Control of Redundant Serial Manipulators." Machines 12, no. 1 (2023): 10. http://dx.doi.org/10.3390/machines12010010.

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Kinematic control of redundant serial manipulators has been carried out for the past half century based primarily on a generalized inverse velocity formulation that is known to have mathematical deficiencies. A recently developed inverse kinematic configuration mapping is employed in an operational configuration space differentiable manifold formulation for redundant-manipulator kinematic control with obstacle avoidance. This formulation is shown to resolve deficiencies in the generalized inverse velocity formulation, especially for high-degree-of-redundancy manipulators. Tracking a specified

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26

Biton, A., E. Rabinovich, and E. Gilad. "Generalized correlation for onset flooding velocity in vertical channels." International Communications in Heat and Mass Transfer 138 (November 2022): 106366. http://dx.doi.org/10.1016/j.icheatmasstransfer.2022.106366.

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27

Derbel, Oussama, Tamás Péter, Benjamin Mourllion, and Michel Basset. "Generalized velocity–density model based on microscopic traffic simulation." Transport 33, no. 2 (2017): 489–501. http://dx.doi.org/10.3846/16484142.2017.1292950.

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In case of the Intelligent Driver Model (IDM) the actual Velocity–Density law V(D) applied by this dynamic system is not defined, only the dynamic behaviour of the vehicles/drivers is determined. Therefore, the logical question is whether the related investigations enhance an existing and known law or reveal a new connection. Specifically, which function class/type is enhanced by the IDM? The publication presents a model analysis, the goal of which was the exploration of a feature of the IDM, which, as yet, ‘remained hidden’. The theoretical model results are useful, this analysis important in

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28

KC, Durga, Dipendra Regmi, Lizheng Tao, and Jiahong Wu. "Generalized 2D Euler–Boussinesq equations with a singular velocity." Journal of Differential Equations 257, no. 1 (2014): 82–108. http://dx.doi.org/10.1016/j.jde.2014.03.012.

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29

Zhang, Chi, Xiangyu Y. Hu, and Nikolaus A. Adams. "A generalized transport-velocity formulation for smoothed particle hydrodynamics." Journal of Computational Physics 337 (May 2017): 216–32. http://dx.doi.org/10.1016/j.jcp.2017.02.016.

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30

Johnson, G. R., S. R. Beissel, and R. A. Stryk. "A generalized particle algorithm for high velocity impact computations." Computational Mechanics 25, no. 2-3 (2000): 245–56. http://dx.doi.org/10.1007/s004660050473.

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31

Zanette, Damián H. "Solitonic solutions for the generalized two-velocity Boltzmann equation." Physica A: Statistical Mechanics and its Applications 153, no. 3 (1988): 612–18. http://dx.doi.org/10.1016/0378-4371(88)90245-2.

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32

Zbrutsky, Oleksander, Volodymyr Gavrylenko, and Viktor Trunov. "Features of dynamics and errors of vibrating gyroscopes." MECHANICS OF GYROSCOPIC SYSTEMS, no. 39 (May 20, 2020): 55–66. http://dx.doi.org/10.20535/0203-3771392020229094.

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The obtained generalized mathematical model of micromechanical and rotary vibrating gyroscopes as meters of angular velocity and transfer functions of meters. The features of the dynamics of vibration gyros according to the generalized model and their influence on the properties of the transfer function of the angular velocity meter are determined. The properties of the constant error and the trend of the output signal of micromechanical gyroscopes are investigated.

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Chen, Xiaochun, and Sadek Gala. "ON THE REGULARITY CRITERIA FOR THE GENERALIZED VISCOUS MHD EQUATIONS." Asian-European Journal of Mathematics 04, no. 03 (2011): 403–11. http://dx.doi.org/10.1142/s1793557111000320.

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In this paper, we consider regularity criteria for solutions to the 3D generalized MHD equations with fractional dissipative term -(-Δ)αu for the velocity field and -(-Δ)βb for the magnetic field. For the case α = β, it is proved that if the velocity field belongs to [Formula: see text] with 0 ≤ r < α or the gradient of velocity field belongs to [Formula: see text] with 0 ≤ r ≤ α on [0, T], then the solution remains smooth on [0, T].

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34

Salim, Ashadi. "Analisis Data Seismik Refraksi dengan Metode Generalized-Reciprocal." ComTech: Computer, Mathematics and Engineering Applications 3, no. 1 (2012): 162. http://dx.doi.org/10.21512/comtech.v3i1.2397.

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The analysis of seismic refraction data by the generalized reciprocal method can be used for delineating undulating refractors. The forward and reverse times of arrival at different geophones with XY distance along a refraction profile, are used for calculating time depth. The seismic wave velocity in refractor may be obtained from velocity analysis function, and the depth of refractor under each geophone is obtained from time-depths function. This method has been applied at one line of seismic refraction measurement that was 440 m long with 45 geophone positions. The measurement obtained 20 m

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Saleh, Azzeldeen A., and John P. Castagna. "Revisiting the Wyllie time average equation in the case of near‐spherical pores." GEOPHYSICS 69, no. 1 (2004): 45–55. http://dx.doi.org/10.1190/1.1649374.

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We suggest a generalized form of the Wyllie equation to better describe acoustic velocity variation with porosity in rocks with a mixture of intercrystalline and near‐spherical pores. Introducing a pore shape factor in the Wyllie equation allows different weights to be applied to the solid‐phase P‐wave velocity to account for rock frame stiffness. This is demonstrated in carbonate and igneous rocks having variable amounts of spherical porosity, ranging from 0% to 80% of the bulk porosity. The generalized equation describes a velocity–porosity envelope instead of the simple line given by the Wy

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HARKO, TIBERIU, and FRANCISCO S. N. LOBO. "GENERALIZED DARK GRAVITY." International Journal of Modern Physics D 21, no. 11 (2012): 1242019. http://dx.doi.org/10.1142/s0218271812420199.

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The late-time cosmic acceleration may be due to infra-red modifications of General Relativity. In particular, we consider a maximal extension of the Hilbert–Einstein action and analyze several interesting features of the theory. Generally, the motion is nongeodesic and takes place in the presence of an extra force, which is orthogonal to the four-velocity. These models could lead to some major differences, as compared to the predictions of General Relativity or other modified theories of gravity, in several problems of current interest, such as cosmology, gravitational collapse or the generati

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Yin, Ze, Yong Jun Jian, Long Chang, Ren Na, and Quan Sheng Liu. "Transient AC Electro-Osmotic Flow of Generalized Maxwell Fluids through Microchannels." Applied Mechanics and Materials 548-549 (April 2014): 216–23. http://dx.doi.org/10.4028/www.scientific.net/amm.548-549.216.

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In this paper, we represent analytical solutions of transient velocity for electroosmotic flow (EOF) of generalized Maxwell fluids through both micro-parallel channel and micro-tube using the method of Laplace transform. We solve the problem including the linearized Poisson-Boltzmann equation, the Cauchy momentum equation and generalized Maxwell constitutive equation. By numerical calculation, the results show that the EOF velocity is greatly depends on oscillating Reynolds number and normalized relaxation time.

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38

Feng, B., W. Xu, R. S. Wu, X. B. Xie, and H. Wang. "Finite-frequency traveltime tomography using the Generalized Rytov approximation." Geophysical Journal International 221, no. 2 (2020): 1412–26. http://dx.doi.org/10.1093/gji/ggaa067.

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SUMMARY Wave-equation-based traveltime tomography has been extensively applied in both global tomography and seismic exploration. Typically, the traveltime Fréchet derivative is obtained using the first-order Born approximation, which is only satisfied for weak velocity perturbations and small phase shifts (i.e. the weak-scattering assumption). Although the small phase-shift restriction can be handled with the Rytov approximation, the weak velocity-perturbation assumption is still a major limitation. The recently developed generalized Rytov approximation (GRA) method can achieve an improved ph

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Rybakov, Yuri P., and Natalya V. Semenova. "Generalized Darcy’s Law in Filtration Theory." EPJ Web of Conferences 173 (2018): 02017. http://dx.doi.org/10.1051/epjconf/201817302017.

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We study the hydrodynamics of flow in a porous medium modeling the grain filling in filters. Using the lattice approximation, we derive the structure of the current in porous media and obtain the transverse diffusion coefficient D which proves to be proportional to the diameter d of the grains as constituents of the medium. We consider the axially-symmetric stationary flow in a cylindrical filter and show that the vertical velocity takes its maximal value at the wall, this effect being known as the “near-wall” one. We analyze the solution to the Euler equation with the modified Darcy force, wh

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Mavko, Gary, Tapan Mukerji, and Nicola Godfrey. "Predicting stress‐induced velocity anisotropy in rocks." GEOPHYSICS 60, no. 4 (1995): 1081–87. http://dx.doi.org/10.1190/1.1443836.

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A simple transformation, using measured isotropic [Formula: see text] and [Formula: see text] versus hydrostatic pressure, is presented for predicting stress‐induced seismic velocity anisotropy in rocks. The compliant, crack‐like portions of the pore space are characterized by generalized compressional and shear compliances that are estimated from the isotropic [Formula: see text] and [Formula: see text]. The physical assumption that the compliant porosity is crack‐like means that the pressure dependence of the generalized compliances is governed primarily by normal tractions resolved across c

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KAPLANSKI, FELIX, SERGEI S. SAZHIN, YASUHIDE FUKUMOTO, STEVEN BEGG, and MORGAN HEIKAL. "A generalized vortex ring model." Journal of Fluid Mechanics 622 (March 10, 2009): 233–58. http://dx.doi.org/10.1017/s0022112008005168.

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A conventional laminar vortex ring model is generalized by assuming that the time dependence of the vortex ring thickness ℓ is given by the relation ℓ = atb, where a is a positive number and 1/4 ≤ b ≤ 1/2. In the case in which $a=\sqrt{2\nu}$, where ν is the laminar kinematic viscosity, and b = 1/2, the predictions of the generalized model are identical with the predictions of the conventional laminar model. In the case of b = 1/4 some of its predictions are similar to the turbulent vortex ring models, assuming that the time-dependent effective turbulent viscosity ν∗ is equal to ℓℓ′. This gene

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Vasilyev, Leonid, and Florin Adrian Radu. "On the ability of the Generalized Continuum Transport Model to properly capture dispersion." Analele Universitatii "Ovidius" Constanta - Seria Matematica 26, no. 2 (2018): 285–99. http://dx.doi.org/10.2478/auom-2018-0029.

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Abstract The ability of the Generalized Continuum Transport model to describe dispersion is studied through the comparison of the breakthrough curves with an analytical solution of the linear advection-dispersion Equation. First, a velocity distribution due to Taylor dispersion in a capillary tube is related to the dispersion coeficient of the advection-dispersion equation. The same distribution is applied to the Generalized Continuum Transport model, where the dispersive flux term is not included as the term proportional to the concentration gradient. In the second stage the velocity distribu

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Bridle, Ralph, and Shelton Hubbell. "Inversion of donor vertical time shifts to update a near-surface depth/velocity model." GEOPHYSICS 80, no. 2 (2015): B23—B33. http://dx.doi.org/10.1190/geo2014-0040.1.

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The near-surface model for static corrections requires a consistent regional depth/velocity model, while incorporating the fidelity of additional static solutions. We addressed the challenge of tying new seismic acquisition to a depth/velocity model, in which there are static corrections derived independently for each seismic data. The generalized method is a four-step procedure that starts with the grafting of the additional shifts to the recipient static model. The time shifts were then adjusted to constrain the long wavelength at defined locations. The next procedure was to split the time s

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Potoczkaya, A. V., A. M. Kovalevskij, V. A. Zheleznyak, and A. A. Komova. "Physiotherapy impact on the periodontal microcirculation during mild chronic generalized periodontitis treatment." Parodontologiya 27, no. 3 (2022): 243–49. http://dx.doi.org/10.33925/1683-3759-2022-27-3-243-249.

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Relevance. The high incidence of chronic generalized periodontitis indicates the relevance of this disease and its social significance. Treatment of chronic generalized periodontitis requires an integrated approach and should include conservative and surgical methods and elements of dental system rehabilitation. Physiotherapy procedures, which affect tissue blood supply and increase local immunity, form an important component of the complex treatment of chronic generalized periodontitis. Choosing the most effective physiotherapy method is essential for the complex treatment of chronic generali

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Ruffa, Anthony A. "The generalized method of exhaustion." International Journal of Mathematics and Mathematical Sciences 31, no. 6 (2002): 345–51. http://dx.doi.org/10.1155/s0161171202011638.

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The method of exhaustion is generalized to a simple integration formula that is valid for the Riemann integrable functions. Both a geometric approach (following the usual procedure for the method of exhaustion) and an independent algebraic generalization approach are provided. Applications provided as examples include use of the formula to generate new series for common functions as well as computing the group velocity distribution resulting from waves diffracted from an aperture.

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Zhang, Z. M., and Keunhan Park. "On the Group Front and Group Velocity in a Dispersive Medium Upon Refraction From a Nondispersive Medium." Journal of Heat Transfer 126, no. 2 (2004): 244–49. http://dx.doi.org/10.1115/1.1668035.

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Conventional definitions of velocities associated with the propagation of modulated waves cannot clearly describe the behavior of the wave packet in a multidimensional dispersive medium. The conventional definition of the phase velocity, which is perpendicular to the wave front, is a special case of the generalized phase velocity defined in this work, since there exist an infinite number of solutions to the equation describing the wave-front movement. Similarly, the generalized group-front velocity is defined for the movement of a wave packet in an arbitrary direction. The group-front velocity

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47

Loduha, T. A., and B. Ravani. "On First-Order Decoupling of Equations of Motion for Constrained Dynamical Systems." Journal of Applied Mechanics 62, no. 1 (1995): 216–22. http://dx.doi.org/10.1115/1.2895905.

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In this paper we present a method for obtaining first-order decoupled equations of motion for multirigid body systems. The inherent flexibility in choosing generalized velocity components as a function of generalized coordinates is used to influence the structure of the resulting dynamical equations. Initially, we describe how a congruency transformation can be formed that represents the transformation between generalized velocity components and generalized coordinate derivatives. It is shown that the proper choice for the congruency transformation will insure generation of first-order decoupl

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48

Ursin, Bjørn, and Alexey Stovas. "Generalized Dix equations for a layered transversely isotropic medium." GEOPHYSICS 70, no. 6 (2005): D77—D81. http://dx.doi.org/10.1190/1.2127111.

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In standard Dix equations, normal-incidence traveltimes and rms or NMO velocities for PP reflections are used to estimate interval velocity and thickness for homogeneous isotropic horizontal layers. For a stack of homogeneous horizontal layers that are transversely isotropic with a vertical symmetry axis (VTI), the qPqP traveltimes (and the qPqSV traveltimes) depend on four parameters and the thickness of each layer. These can be estimated from the traveltimes of the qPqP and qPqSV reflected waves. The estimated qPqP traveltime parameters are P-wave normal-incidence traveltime, P-wave NMO velo

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49

Viúdez, Álvaro. "The Vorticity–Velocity Gradient Cofactor Tensor and the Material Invariant of the Semigeostrophic Theory." Journal of the Atmospheric Sciences 62, no. 7 (2005): 2294–301. http://dx.doi.org/10.1175/jas3482.1.

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Abstract A new derivation and interpretation of the semigeostrophic (SG) material invariant in the theory of geophysical flows is introduced. First, a generalized three-dimensional equation of the SG dynamics is established and the generalized equations for the rate of change of vorticity and for the rate of change of the velocity gradient cofactor tensor are obtained. Next, a conservation equation for the vorticity–velocity gradient cofactor tensor (denoted Ξ̃) is derived. The specific potential Ξ̃, that is, Ξ̃ in the reference configuration per unit of mass, is defined and an expression for

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Kholodov, Yaroslav, Andrey Alekseenko, Viktor Kazorin, and Alexander Kurzhanskiy. "Generalization Second Order Macroscopic Traffic Models via Relative Velocity of the Congestion Propagation." Mathematics 9, no. 16 (2021): 2001. http://dx.doi.org/10.3390/math9162001.

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This paper presents a generalized second-order hydrodynamic traffic model. Its central piece is the expression for the relative velocity of the congestion (compression wave) propagation. We show that the well-known second-order models of Payne–Whitham, Aw–Rascal and Zhang are all special cases of the featured generalized model, and their properties are fully defined by how the relative velocity of the congestion is expressed. The proposed model is verified with traffic data from a segment of the Interstate 580 freeway in California, USA, collected by the California DOT’s Performance Measuremen

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