scholarly journals GEODYNAMICS

GEODYNAMICS ◽  
2021 ◽  
Vol 2(31)2021 (2(31)) ◽  
pp. 29-40
Author(s):  
Mykhailo Fys ◽  
◽  
Andrii Brydun ◽  
Mariana Yurkiv ◽  
Andrii Sohor ◽  
...  
Keyword(s):  
Distribution Function ◽  
Mass Distribution ◽  
Dynamic Compression ◽  
Center Of Mass ◽  
Three Dimensional ◽  
The Earth ◽  
The Mean ◽  
Mass Distribution Function ◽  
The Given ◽  
Derivatives Of

Purpose. To investigate the features of the algorithm implementation for finding the derivatives of the spatial distribution function of the planet's masses with the use of high-order Stokes constants and, on the basis of this, to find its analytical expression. According to the given methodology, to carry out calculations with the help of which to carry on the study of dynamic phenomena occurring inside an ellipsoidal planet. The proposed method involves the determination of the derivatives of the mass distribution function by the sum, the coefficients of which are obtained from the system of equations, which is incorrect. In order to solve it, an error-resistant method for calculating unknowns was used. The implementation of the construction is carried out in an iterative way, while for the initial approximation we take the three-dimensional function of the density of the Earth's masses, built according to Stokes constants up to the second order inclusive, by dynamic compression by the one-dimensional density distribution, and we determine the expansion coefficients of the derivatives of the function in the variables to the third order inclusive. They are followed by the corresponding density function, which is then taken as the initial one. The process is repeated until the specified order of approximation is reached. To obtain a stable result, we use the Cesaro summation method (method of means).. The calculations performed with the help of programs that implement the given algorithm, while the achieved high (ninth) order of obtaining the terms of the sum of calculations. The studies of the convergence of the sum of the series have been carried out, and on this basis, a conclusion has been made about the advisability of using the generalized finding of the sums based on the Cesaro method. The optimal number of contents of the sum terms has been chosen, provides convergence both for the mass distribution function and for its derivatives. Calculations of the deviations of mass distribution from the mean value ("inhomogeneities") for extreme points of the earth's geoid, which basically show the total compensation along the radius of the Earth, have been performed. For such three-dimensional distributions, calculations were performed and schematic maps were constructed according to the taken into account values of deviations of three-dimensional distributions of the mean ("inhomogeneities") at different depths reflecting the general structure of the Earth's internal structure. The presented vector diagrams of the horizontal components of the density gradient at characteristic depths (2891 km - core-mantle, 700 km - middle of the mantle, also the upper mantle - 200, 100 km) allow us to draw preliminary conclusions about the global movement of masses. At the same time, a closed loop is observed on the “core-mantle” edge, which is an analogy of a closed electric circuit. For shallower depths, differentiation of vector motions is already taking place, which gives hope for attracting these vector-grams to the study of dynamic motions inside the Earth. In fact, the vertical component (derivative with respect to the z variable) is directed towards the center of mass and confirms the main property of mass distributions - growth when approaching the center of mass. The method of stable solution of incorrect linear systems is applied, by means of which the vector-gram of the gradient of the mass distribution function is constructed. The nature of such schemes provides a tool for possible causes of mass redistribution in the middle of the planet and to identify possible factors of tectonic processes in the middle of the Earth, i.e indirectly confirms the gravitational convection of masses. The proposed technique can be used to create detailed models of density functions and its characteristics (derivatives) of the planet's interior, and the results of numerical experiments - to solve tectonics problems.

scholarly journals 4. Factors affecting radar-meteor echo durations

1968 ◽  
Vol 33 ◽  
pp. 45-49
Author(s):  
A. Hajduk
Keyword(s):  
Distribution Function ◽  
Mass Distribution ◽  
Meteor Shower ◽  
Time Intervals ◽  
Factors Affecting ◽  
Factors Influencing ◽  
Two Factors ◽  
The Mean ◽  
Mass Distribution Function ◽  
Radar Meteor

The analysis of two factors influencing radar-meteor echo durations leads to the conclusion, that (1) the echo duration depends considerably on the train position with respect to the sensitivity contours of the radar; (2) the mean echo duration changes with respect to the radiant motion of a meteor shower. As a consequence of the factors mentioned above, the magnitude function, or the mass distribution function depends significantly on the observational conditions, as well as on the choice of the range and time intervals of the investigated sample.

THREE DIMENSIONAL INTERPRETATION OF GRAVITATIONAL ANOMALIES

Geophysics ◽  
1952 ◽  
Vol 17 (2) ◽  
pp. 344-364 ◽  
Author(s):  
Fraser S. Grant
Keyword(s):  
Gravitational Field ◽  
Mass Distribution ◽  
Three Dimensional ◽  
Relaxation Method ◽  
The Body ◽  
Multipole Moments ◽  
Size And Shape ◽  
Surface Field ◽  
Derivatives Of

A method is developed for determining the approximate size and shape of the three‐dimensional mass distribution that is required to produce a given gravitational field. The first few reduced multipole moments of the distribution are calculated from the derivatives of the surface field, and the approximative structure is determined from the values of these moments and a knowledge of the density contrast between the body and its surroundings. A system of classification of problems by symmetry is introduced and its practical usage discussed. A relaxation method is described which may be used to adjust the initial solution systematically to give agreement over the whole field. A descriptive discussion is appended.

On approach to determine the internal potential and gravitational energy of ellipsoid

2021 ◽  
Vol 8 (3) ◽  
pp. 359-367
Author(s):  
М. M. Fys ◽  
◽  
А. M. Brydun ◽  
М. I. Yurkiv ◽  
◽  
...  
Keyword(s):  
Distribution Function ◽  
Mass Distribution ◽  
Special Form ◽  
Iterative Process ◽  
Three Dimensional ◽  
Gravitational Energy ◽  
One Dimensional ◽  
Piecewise Continuous ◽  
The Masses ◽  
Internal Potential

Formulas are derived for the calculation of the potential of bodies, which surface is a sphere or an ellipsoid, and the distribution function has a special form: a piecewise continuous one-dimensional function and a three-dimensional mass distribution. For each of these cases, formulas to calculate both external and internal potentials are derived. With their help, further the expressions are given for calculation of the potential (gravitational) energy of the masses of such bodies and their corresponding distributions. For spherical bodies, the exact and approximate relations for determining the energy are provided, which makes it possible to compare the iterative process and the possibility of its application to an ellipsoid. The described technique has been tested by a specific numerical example.

scholarly journals Numerical Modeling of Force-Stiffness Response of Cross-Linked Actin Networks Using Tensegrity Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Xian Xu ◽  
Yafeng Wang ◽  
Yaozhi Luo
Keyword(s):  
Three Dimensional ◽  
Structural Response ◽  
Postbuckling Behavior ◽  
Actin Networks ◽  
Tensegrity Structure ◽  
Tensegrity Systems ◽  
Order Polynomial ◽  
The Mean ◽  
The Given ◽  
Tensegrity Model

A three-dimensional tensegrity structure is used as a computational model for cross-linked actin networks. The postbuckling behavior of the members under compression is considered and the constitutive relation of the postbuckling members is modeled as a second-order polynomial. A numerical scheme incorporating the equivalent constitution of the postbuckling members is used to predict the structural response of the tensegrity model under compression loads. The numerical simulation shows that the stiffness of the tensegrity structure nonlinearly increases before member buckling and abruptly decreases to a lower level as soon as members buckle. This result qualitatively mimics the experimentally observed stiffness to compression stress response of cross-linked actin networks. In order to take member length variety into account, a large number of simulations with the length of buckling members varying in the given range are also carried out. It is found that the mean response of the simulations using different buckling member length exhibits more resemblance to the experimental observation.

The mass distribution function of groups of galaxies

1992 ◽  
Vol 389 ◽  
pp. 68 ◽  
Author(s):  
A. Pisani ◽  
G. Giuricin ◽  
F. Mardirossian ◽  
M. Mezzetti
Keyword(s):  
Distribution Function ◽  
Mass Distribution ◽  
Groups Of Galaxies ◽  
Mass Distribution Function

scholarly journals Some Problems Connected with the Calculation of Stop Loss Premiums for Large Portfolios

1967 ◽  
Vol 4 (2) ◽  
pp. 170-174 ◽  
Author(s):  
Fredrik Esscher
Keyword(s):  
Distribution Function ◽  
Expected Number ◽  
Limit Values ◽  
Empirical Determination ◽  
The Mean ◽  
Premium Rates ◽  
Image Position ◽  
Stop Loss ◽  
The Given

When experience is insufficient to permit a direct empirical determination of the premium rates of a Stop Loss Cover, we have to fall back upon mathematical models from the theory of probability—especially the collective theory of risk—and upon such assumptions as may be considered reasonable.The paper deals with some problems connected with such calculations of Stop Loss premiums for a portfolio consisting of non-life insurances. The portfolio was so large that the values of the premium rates and other quantities required could be approximated by their limit values, obtained according to theory when the expected number of claims tends to infinity.The calculations were based on the following assumptions.Let F(x, t) denote the probability that the total amount of claims paid during a given period of time is ≤ x when the expected number of claims during the same period increases from o to t. The net premium II (x, t) for a Stop Loss reinsurance covering the amount by which the total amount of claims paid during this period may exceed x, is defined by the formula and the variance of the amount (z—x) to be paid on account of the Stop Loss Cover, by the formula As to the distribution function F(x, t) it is assumed that wherePn(t) is the probability that n claims have occurred during the given period, when the expected number of claims increases from o to t,V(x) is the distribution function of the claims, giving the conditioned probability that the amount of a claim is ≤ x when it is known that a claim has occurred, andVn*(x) is the nth convolution of the function V(x) with itself.V(x) is supposed to be normalized so that the mean = I.

Estimation of Balance Stability States for Legged Robotic Systems

2016 ◽  
Author(s):  
Carlotta Mummolo ◽  
Luigi Mangialardi ◽  
Joo H. Kim
Keyword(s):  
State Space ◽  
Degrees Of Freedom ◽  
Center Of Mass ◽  
Stability Boundary ◽  
Three Dimensional ◽  
Quantitative Measure ◽  
Initial Position ◽  
Robotic Systems ◽  
The Given ◽  
Balance Stability

A three-dimensional criterion is provided for the estimation of balance stability states of legged robotic systems subject to various constraints. A general framework is established to evaluate the balance stability boundary of a given system in the state space of Center of Mass (COM) Cartesian position and velocity. For each assigned COM initial position, an optimization-based iterative algorithm finds the minimum and maximum COM initial velocity that the system can handle along a given direction, such that it maintains the capability to reach a final static equilibrium. The resulting set of velocity extrema constitutes the system’s balance stability boundary, which represents the sufficient condition to estimate falling states versus balanced states, according to the definitions provided herein. The COM state space domain identified with this approach contains all possible balanced states for the given legged system, with respect to the necessary physical, balancing, and design constraints. The balance state estimation is demonstrated for 1- and 2-degrees of freedom planar legged systems in single support. The domain identified by the balance stability boundary can be used as a “map” for the given legged system in which the distance from a given state to the domain boundaries can provide a quantitative measure of balance stability/instability.

scholarly journals Bivariate luminosity-HI mass distribution function of galaxies based on the NIBLES survey

2018 ◽  
Vol 619 ◽  
pp. A89 ◽  
Author(s):  
Z. Butcher ◽  
S. Schneider ◽  
W. van Driel ◽  
M. D. Lehnert
Keyword(s):  
Mass Distribution ◽  
Luminosity Function ◽  
Hubble Constant ◽  
Mass Function ◽  
Volume Density ◽  
Local Universe ◽  
Mass Distributions ◽  
Low Mass ◽  
The Mean ◽  
Mass Distribution Function

We present a new optical luminosity-HI mass bivariate luminosity function (BLF) based on HI line observations from the Nançay Interstellar Baryons Legacy Extragalactic Survey (NIBLES). NIBLES sources lie within the local universe (900 ≤ c z ≤ 12 000 km s−1) and were chosen from SDSS DR5 such that the optical luminosity function was sampled as uniformly as possible. The HI mass function (HIMF) derived from our raw-data BLF, which is based on HI detections only, is consistent with the HIMFs derived from other optically selected surveys in that the low-mass slope is flatter than those derived from blind HI surveys. However, spanning the entire luminosity range of NIBLES, we identify a highly consistent distribution of the HI gas mass to luminosity ratio (gas-to-light ratio) with a predictable progression in the mean MHI/L r ratio as a function of L r. This consistency allows us to construct plausible gas-to-light ratio distributions for very low-luminosity bins which lie outside the NIBLES sample. We also identify a ∼10% decrease in detection fraction for galaxies fainter than log(L r) = 9.25, consistent with the expected decrease due to distance and sensitivity effects. Accounting for these trends, we reconstruct plausible gas-to-light distributions spanning luminosity bins down to log(L r) = 5.25, thus producing a corrected BLF. This corrected BLF is in good qualitative agreement with optical luminosity-HI mass distributions from the ALFALFA survey and is able to accurately reproduce blind survey HIMFs, lending credibility that this two dimensional optical luminosity-HI mass distribution is an accurate representation of the volume density distribution of galaxies in the local universe. We also note that our agreement with HIMFs from other surveys is dependent on accounting for all systematic differences such as selection method, Hubble constant and HI flux scale.

Material and Local Time Derivatives of Screws With Applications to Dynamics and Stiffness

2004 ◽  
Author(s):  
Harvey Lipkin
Keyword(s):  
Elastic System ◽  
Center Of Mass ◽  
Three Dimensional ◽  
Rigid Bodies ◽  
Material Derivative ◽  
Gravitational Loading ◽  
Moving Body ◽  
Rigid Body Mechanics ◽  
Derivatives Of ◽  
Insight Into

Screw quantities provide geometric insight into three-dimensional mechanics modeled by rigid bodies and lumped parameters. Four distinct cases of time differentiation are examined by combining fixed and moving body derivatives (fundamental to rigid body mechanics) with material and local derivatives (fundamental to continuum mechanics). Three combinations always yield another screw quantity while the most common, the material derivative with respect to the fixed body, does not. Two fundamental formulations are examined with this last derivative, Euler’s Laws and the gravitational loading of an elastic system. By coincidence, the formulations appear screw-like when they are expressed at the center-of-mass but, in contrast to actual screw formulations, they do not retain invariant forms when expressed at arbitrary points.

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