Calibration and Correction Method of the Deflection Angle of Rotation Axis Projection on Neutron Tomography

To better evaluate the spatial steering effect of directional perforation hydraulic fractures, evaluation indexes for the spatial steering effect are first proposed in this paper. Then, these indexes are used to quantitatively evaluate existing physical experimental results. Finally, with the help of RFPA2D-Flow software, the influence of perforation length and azimuth on the spatial steering process of hydraulic fracture are quantitatively analysed using four evaluation indexes. It is shown by the results that the spatial deflection trajectory, deflection distance, deflection angle and initiation pressure of hydraulic fractures can be used as quantitative evaluation indexes for the spatial steering effect of hydraulic fractures. The deflection paths of directional perforation hydraulic fractures are basically the same. They all gradually deflect to the maximum horizontal principal stress direction from the perforation hole and finally represent a double-wing bending fracture. The deflection distance, deflection angle and initiation pressure of hydraulic fractures increase gradually with increasing perforation azimuth, and the sensitivity of the deflection angle to the perforation azimuth of hydraulic fractures also increases. With increasing perforation length, the deflection distance of hydraulic fractures increases gradually. However, the deflection angle and initiation pressure decrease gradually, as does the sensitivity.

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OPERATOR-PRODUCT EXPANSION AND ANALYTICITY

International Journal of Modern Physics A ◽

10.1142/s0217751x9900227x ◽

1999 ◽

Vol 14 (30) ◽

pp. 4819-4840

Author(s):

JAN FISCHER ◽

IVO VRKOČ

Keyword(s):

Operator Product Expansion ◽

Deflection Angle ◽

Vacuum Expectation ◽

Vacuum Expectation Value ◽

Operator Product ◽

Coupling Constant ◽

Expectation Value ◽

Euclidean Region ◽

The Deflection Angle ◽

Class Of Functions

We discuss the current use of the operator-product expansion in QCD calculations. Treating the OPE as an expansion in inverse powers of an energy-squared variable (with possible exponential terms added), approximating the vacuum expectation value of the operator product by several terms and assuming a bound on the remainder along the Euclidean region, we observe how the bound varies with increasing deflection from the Euclidean ray down to the cut (Minkowski region). We argue that the assumption that the remainder is constant for all angles in the cut complex plane down to the Minkowski region is not justified. Making specific assumptions on the properties of the expanded function, we obtain bounds on the remainder in explicit form and show that they are very sensitive both to the deflection angle and to the class of functions considered. The results obtained are discussed in connection with calculations of the coupling constant αs from the τ decay.

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The precise calculations of the constant terms in the equations of motions of planets and photons of general relativity

Physics Essays ◽

10.4006/0836-1398-34.2.183 ◽

2021 ◽

Vol 34 (2) ◽

pp. 183-192

Author(s):

Mei Xiaochun

Keyword(s):

General Relativity ◽

Gravitational Field ◽

Deflection Angle ◽

Constant Term ◽

Equation Of Motion ◽

Time Dependent ◽

Newtonian Gravity ◽

Newtonian Theory ◽

The Deflection Angle ◽

Equations Of Motions

In general relativity, the values of constant terms in the equations of motions of planets and light have not been seriously discussed. Based on the Schwarzschild metric and the geodesic equations of the Riemann geometry, it is proved in this paper that the constant term in the time-dependent equation of motion of planet in general relativity must be equal to zero. Otherwise, when the correction term of general relativity is ignored, the resulting Newtonian gravity formula would change its basic form. Due to the absence of this constant term, the equation of motion cannot describe the elliptical and the hyperbolic orbital motions of celestial bodies in the solar gravitational field. It can only describe the parabolic orbital motion (with minor corrections). Therefore, it becomes meaningless to use general relativity calculating the precession of Mercury's perihelion. It is also proved that the time-dependent orbital equation of light in general relativity is contradictory to the time-independent equation of light. Using the time-independent orbital equation to do calculation, the deflection angle of light in the solar gravitational field is <mml:math display="inline"> <mml:mrow> <mml:mn>1.7</mml:mn> <mml:msup> <mml:mn>5</mml:mn> <mml:mo>″</mml:mo> </mml:msup> </mml:mrow> </mml:math> . But using the time-dependent equation to do calculation, the deflection angle of light is only a small correction of the prediction value <mml:math display="inline"> <mml:mrow> <mml:mn>0.87</mml:mn> <mml:msup> <mml:mn>5</mml:mn> <mml:mo>″</mml:mo> </mml:msup> </mml:mrow> </mml:math> of the Newtonian gravity theory with a magnitude order of <mml:math display="inline"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mn>10</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . The reason causing this inconsistency was the Einstein's assumption that the motion of light satisfied the condition <mml:math display="inline"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> in gravitational field. It leads to the absence of constant term in the time-independent equation of motion of light and destroys the uniqueness of geodesic in curved space-time. Meanwhile, light is subjected to repulsive forces in the gravitational field, rather than attractive forces. The direction of deflection of light is opposite, inconsistent with the predictions of present general relativity and the Newtonian theory of gravity. Observing on the earth surface, the wavelength of light emitted by the sun is violet shifted. This prediction is obviously not true. Practical observation is red shift. Finally, the practical significance of the calculation of the Mercury perihelion's precession and the existing problems of the light's deflection experiments of general relativity are briefly discussed. The conclusion of this paper is that general relativity cannot have consistence with the Newtonian theory of gravity for the descriptions of motions of planets and light in the solar system. The theory itself is not self-consistent too.

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Gauging the earth

The Mathematical Gazette ◽

10.1017/s0025557200000140 ◽

2013 ◽

Vol 97 (540) ◽

pp. 413-420

Author(s):

John D. Mahony

Keyword(s):

Approximate Formula ◽

Deflection Angle ◽

School Science ◽

Science Department ◽

Measurement Apparatus ◽

The Earth ◽

Interesting Discussion ◽

Simple Measurement ◽

The Deflection Angle

In an earlier note by W. J. A. Colman [1], the reader was treated to an interesting discussion concerning a BBC programme about the earth's radius in the context of endeavours to determine it by an 11th century Persian mathematician, Al-Biruni. At the same time a modem but approximate formula that might not have been available to the ancients was proposed for the radius and, following that account, it is the purpose here to explore just how accurately one might determine the earth's radius using this formula together with a simple measurement apparatus (not a sophisticated astrolabe) that might be constructed from materials found in the garage of any DIY handyman, or indeed, in the laboratory of any school science department. A schematic for the configuration of an observer P at a height h above the earth (of radius R) where the deflection angle to the horizon is denoted by θ, is shown in Figure 1. Also shown in the figure is a ‘sighting tube’ of length L about which more will be said later.

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SPECIAL VEHICLE MOTION STABILIZATION SYSTEM

IZVESTIA VOLGOGRAD STATE TECHNICAL UNIVERSITY ◽

10.35211/1990-5297-2021-9-256-37-41 ◽

2021 ◽

pp. 37-41

Author(s):

M. V. Lyashenko ◽

V. V. Shekhovtsov ◽

P. V. Potapov ◽

A. A. Shvedunenko

Keyword(s):

Pid Controller ◽

Deflection Angle ◽

Angular Displacement ◽

Dynamic Equations ◽

Engine Torque ◽

Special Vehicle ◽

Vehicle Motion ◽

Motion Stabilization ◽

The Deflection Angle ◽

The Mathematical Model

The system of special vehicle (SV) motion stabilization during moving on a straight surface is modeled on the base of dynamic equations of the mathematical model. The movement is stabilized by using a PID controller, the angular displacement of the mass is selected to ensure a given speed of movement, and the deflection angle is stabilized by controlling the engine torque.

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Values estimation of deflection angle of a floating sensor for various objects (to the resistance to measuring the water surface rate)

Bulletin of Science and Practice ◽

10.33619/2414-2948/39/25 ◽

2019 ◽

Vol 5 (2) ◽

pp. 181-196

Author(s):

K. Presnyakov ◽

G. Kerimkulova ◽

G. Askalieva

Keyword(s):

Water Surface ◽

Deflection Angle ◽

Surface Velocity ◽

Informative Parameter ◽

The Deflection Angle

A device is proposed for measuring the surface velocity of water, the use of which is based on the use of a new informative parameter — the deflection angle of the movable float sensor (together with the guide slide) from the vertical in the direction of the dynamic axis of the flow, allowing you to simply and reliably measure the surface velocity of water.

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Weak Deflection Angle of some Classes of non-linear Electrodynamics Black Holes via Gauss-Bonnet Theorem

10.20944/preprints202008.0370.v1 ◽

2020 ◽

Author(s):

Hasan El Moumni ◽

Karima Masmar ◽

Ali Övgün

Keyword(s):

Black Holes ◽

Black Hole ◽

Gravitational Lensing ◽

Deflection Angle ◽

Weak Field ◽

Leading Order ◽

Plasma Medium ◽

Non Linear ◽

The Deflection Angle ◽

Bonnet Theorem

In this paper, we study the gravitational lensing by some black hole classes within the non-linear electrodynamics in weak field limits. First, we calculate an optical geometry of the non-linear electrodynamics black hole then we use the Gauss-Bonnet theorem for finding deflection angle in weak field limits. The effect of non-linear electrodynamics on the deflection angle in leading order terms is studied. Furthermore, we discuss the effects of the plasma medium on the weak deflection angle.

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Perturbative deflection angle, gravitational lensing in the strong field limit and the black hole shadow

The European Physical Journal C ◽

10.1140/epjc/s10052-021-09026-7 ◽

2021 ◽

Vol 81 (3) ◽

Author(s):

Junji Jia ◽

Ke Huang

Keyword(s):

Gravitational Lensing ◽

Deflection Angle ◽

Strong Field ◽

Distance Effect ◽

Critical Value ◽

Field Limit ◽

Perturbative Method ◽

The Deflection Angle ◽

The Impact ◽

Strong Field Limit

AbstractA perturbative method to compute the deflection angle of both timelike and null rays in arbitrary static and spherically symmetric spacetimes in the strong field limit is proposed. The result takes a quasi-series form of $$(1-b_c/b)$$ ( 1 - b c / b ) where b is the impact parameter and $$b_c$$ b c is its critical value, with coefficients of the series explicitly given. This result also naturally takes into account the finite distance effect of both the source and detector, and allows to solve the apparent angles of the relativistic images in a more precise way. From this, the BH angular shadow size is expressed as a simple formula containing metric functions and particle/photon sphere radius. The magnification of the relativistic images were shown to diverge at different values of the source-detector angular coordinate difference, depending on the relation between the source and detector distance from the lens. To verify all these results, we then applied them to the Hayward BH spacetime, concentrating on the effects of its charge parameter l and the asymptotic velocity v of the signal. The BH shadow size were found to decrease slightly as l increases to its critical value, and increase as v decreases from light speed. For the deflection angle and the magnification of the images however, both the increase of l and decrease of v will increase their values.

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FLUCTUATIONS OF THE CARGO TRANSPORTED BY LIFTING CRANE: SIMULATION AND ANALYSIS

The Russian Automobile and Highway Industry Journal ◽

10.26518/2071-7296-2019-5-526-533 ◽

2019 ◽

Vol 16 (5) ◽

pp. 526-533

Author(s):

M. S. Korytov ◽

V. S. Shcherbakov ◽

V. E. Belyakov

Keyword(s):

Objective Function ◽

Arbitrary Order ◽

Deflection Angle ◽

Suspension Point ◽

Program Code ◽

First Derivative ◽

Urgent Task ◽

Acceleration And Deceleration ◽

The Deflection Angle ◽

Simulation Results

Introduction. Reducing fluctuations in the load transported by hoisting cranes with a flexible rope suspension of the load is an urgent task since it can significantly reduce the time taken to complete the operation of moving the load. A promising direction for reducing load fluctuations is to optimize the trajectory of movement of the load suspension upper point.Materials and methods. The paper discussed the method of mathematical simulation of plane vibrations of a load moved by a crane with a horizontally moving suspension point, using the software of the MATLAB system. For modeling, the authors used the function of the MATLAB ode45 system, intended for the numerical solution of systems of non-stationary differential equations of arbitrary order.The second-order differential equation used to describe the fluctuations of the transported load and its implementation in the form of program code was presented. Moreover, the authors demonstrated the elements of program code for the analysis and visualization of simulation results.Results. The authors obtained and presented the series of graphs in the inclination angle’s changing of the cargo rope, the acceleration of the suspension point and the value of the objective function with the sinusoidal nature of the acceleration. The objective function was the sum of the absolute values of the deflection angle of the rope and the first derivative at the final moment of the suspension point’s movement with acceleration.Discussion and conclusions. As a result, the paper shows that the system with energy dissipation does not reach the zero value of the objective function even by a symmetrical nature of acceleration and deceleration of the suspension point. Therefore, it is necessary to give asymmetry to the acceleration and deceleration periods of the suspension point in order to completely absorb the residual fluctuations of the load.

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Preferentialness Design of a Composite Impeller with Ultra-Low Unit Speed Centrifugal Pump

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.308-310.2353 ◽

2011 ◽

Vol 308-310 ◽

pp. 2353-2357 ◽

Cited By ~ 1

Author(s):

Ya Bin Tian ◽

Xue Yi Qi ◽

Jia Xin Hu

Keyword(s):

Centrifugal Pump ◽

Work Performance ◽

Deflection Angle ◽

Blade Design ◽

Orthogonal Method ◽

Unit Speed ◽

Curve Method ◽

Main Factors ◽

The Deflection Angle ◽

Large Discharge

According to main factors which affect the design of a composite impeller with ultra-low Unit speed centrifugal pump—the leaf number of the composite impeller’s short blade design, the relative position of the blade’s radial inlet, the bias degree of the blade’s circumferential direction, the deflection angle and so on. The paper applies Orthogonal method and establishes 16 design schemes. The preferentialness design of Composite Impeller’s short blade with ultra-low unit speed centrifugal pump was gained by numerical simulation. Contrast to the method of permutation and combination, this kind of method can decrease the designer’s workload greatly. As the results show, the composite impeller’s inner stress field and velocity field distribute reasonably, H - Q curve method is smoother, curve directs to large discharge and the work performance becomes well by applying orthogonal method.

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