scholarly journals Local Fractional Metric Dimensions of Rotationally Symmetric and Planar Networks

IEEE Access ◽  
2020 ◽  
Vol 8 ◽  
pp. 82404-82420 ◽  
Author(s):  
Jia-Bao Liu ◽  
Muhammad Kamran Aslam ◽  
Muhammad Javaid
Keyword(s):  
Rotationally Symmetric ◽  
Metric Dimensions ◽  
Planar Networks

scholarly journals Classification of Upper Bound Sequences of Local Fractional Metric Dimension of Rotationally Symmetric Hexagonal Planar Networks

2021 ◽  
Vol 2021 ◽  
pp. 1-24
Author(s):  
Shahbaz Ali ◽  
Muhammad Khalid Mahmood ◽  
Fairouz Tchier ◽  
F. M. O. Tawfiq
Keyword(s):  
Image Processing ◽  
Chemical Engineering ◽  
Chemical Compounds ◽  
Vital Role ◽  
Metric Dimension ◽  
Integer Programming Problem ◽  
Rotationally Symmetric ◽  
Metric Dimensions ◽  
Planar Networks

The term metric or distance of a graph plays a vital role in the study to check the structural properties of the networks such as complexity, modularity, centrality, accessibility, connectivity, robustness, clustering, and vulnerability. In particular, various metrics or distance-based dimensions of different kinds of networks are used to resolve the problems in different strata such as in security to find a suitable place for fixing sensors for security purposes. In the field of computer science, metric dimensions are most useful in aspects such as image processing, navigation, pattern recognition, and integer programming problem. Also, metric dimensions play a vital role in the field of chemical engineering, for example, the problem of drug discovery and the formation of different chemical compounds are resolved by means of some suitable metric dimension algorithm. In this paper, we take rotationally symmetric and hexagonal planar networks with all possible faces. We find the sequences of the local fractional metric dimensions of proposed rotationally symmetric and planar networks. Also, we discuss the boundedness of sequences of local fractional metric dimensions. Moreover, we summarize the sequences of local fractional metric dimension by means of their graphs.

scholarly journals On the Upper Bounds of Fractional Metric Dimension of Symmetric Networks

2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
Muhammad Javaid ◽  
Muhammad Kamran Aslam ◽  
Jia-Bao Liu
Keyword(s):  
Image Processing ◽  
Drug Discovery ◽  
Computer Science ◽  
Chemical Compounds ◽  
Upper Bounds ◽  
Metric Dimension ◽  
Rotationally Symmetric ◽  
Metric Dimensions ◽  
Structural Aspects ◽  
Symmetric Networks

Distance-based numeric parameters play a pivotal role in studying the structural aspects of networks which include connectivity, accessibility, centrality, clustering modularity, complexity, vulnerability, and robustness. Several tools like these also help to resolve the issues faced by the different branches of computer science and chemistry, namely, navigation, image processing, biometry, drug discovery, and similarities in chemical compounds. For this purpose, in this article, we are considering a family of networks that exhibits rotationally symmetric behaviour known as circular ladders consisting of triangular, quadrangular, and pentagonal faced ladders. We evaluate their upper bounds of fractional metric dimensions of the aforementioned networks.

Practical Correction of Three Fold Astigmatism in the Philips CM TEM

1996 ◽  
Vol 54 ◽  
pp. 418-419
Author(s):  
Arno J. Bleeker ◽  
Mark H.F. Overwijk ◽  
Max T. Otten
Keyword(s):  
Optical Properties ◽  
Phase Contrast ◽  
The Other ◽  
Objective Lens ◽  
Symmetric Part ◽  
A Posteriori ◽  
Contrast Transfer Function ◽  
High Brightness ◽  
Rotationally Symmetric ◽  
Brightness Field

With the improvement of the optical properties of the modern TEM objective lenses the point resolution is pushed beyond 0.2 nm. The objective lens of the CM300 UltraTwin combines a Cs of 0. 65 mm with a Cc of 1.4 mm. At 300 kV this results in a point resolution of 0.17 nm. Together with a high-brightness field-emission gun with an energy spread of 0.8 eV the information limit is pushed down to 0.1 nm. The rotationally symmetric part of the phase contrast transfer function (pctf), whose first zero at Scherzer focus determines the point resolution, is mainly determined by the Cs and defocus. Apart from the rotationally symmetric part there is also the non-rotationally symmetric part of the pctf. Here the main contributors are not only two-fold astigmatism and beam tilt but also three-fold astigmatism. The two-fold astigmatism together with the beam tilt can be corrected in a straight-forward way using the coma-free alignment and the objective stigmator. However, this only works well when the coefficient of three-fold astigmatism is negligible compared to the other aberration coefficients. Unfortunately this is not generally the case with the modern high-resolution objective lenses. Measurements done at a CM300 SuperTwin FEG showed a three fold-astigmatism of 1100 nm which is consistent with measurements done by others. A three-fold astigmatism of 1000 nm already sinificantly influences the image at a spatial frequency corresponding to 0.2 nm which is even above the point resolution of the objective lens. In principle it is possible to correct for the three-fold astigmatism a posteriori when through-focus series are taken or when off-axis holography is employed. This is, however not possible for single images. The only possibility is then to correct for the three-fold astigmatism in the microscope by the addition of a hexapole corrector near the objective lens.

New Feasible Concepts of Aberration Correction for Realizing High-Resolution Electron Microscopes

1990 ◽  
Vol 48 (1) ◽  
pp. 202-203
Author(s):  
H. Rose
Keyword(s):  
Spherical Aberration ◽  
Wave Length ◽  
Electron Wave ◽  
Optical Lens ◽  
Resolution Electron ◽  
Rotationally Symmetric ◽  
Electron Microscopes ◽  
The Cost ◽  
Imaging Performance ◽  
Acceleration Voltage

The imaging performance of the light optical lens systems has reached such a degree of perfection that nowadays numerical apertures of about 1 can be utilized. Compared to this state of development the objective lenses of electron microscopes are rather poor allowing at most usable apertures somewhat smaller than 10-2 . This severe shortcoming is due to the unavoidable axial chromatic and spherical aberration of rotationally symmetric electron lenses employed so far in all electron microscopes.The resolution of such electron microscopes can only be improved by increasing the accelerating voltage which shortens the electron wave length. Unfortunately, this procedure is rather ineffective because the achievable gain in resolution is only proportional to λ1/4 for a fixed magnetic field strength determined by the magnetic saturation of the pole pieces. Moreover, increasing the acceleration voltage results in deleterious knock-on processes and in extreme difficulties to stabilize the high voltage. Last not least the cost increase exponentially with voltage.

scholarly journals Mixed fluid cosmological model in f(R, T) gravity

2020 ◽  
Vol 98 (11) ◽  
pp. 1015-1022 ◽  
Author(s):  
Parbati Sahoo ◽  
Barkha Taori ◽  
K.L. Mahanta
Keyword(s):  
Cosmological Model ◽  
Bianchi Type ◽  
Type I ◽  
Time Varying ◽  
Eos Parameter ◽  
Barotropic Fluid ◽  
Rotationally Symmetric ◽  
Expansion Of The Universe ◽  
The Universe ◽  
Mixed Fluid

We construct a locally rotationally symmetric (LRS) Bianchi type-I cosmological model in f(R, T) theory of gravity when the source of gravitation is a mixture of barotropic fluid and dark energy (DE) by employing a time-varying deceleration parameter. We observe through the behavior of the state finder parameters (r, s) that our model begins from the Einstein static era and goes to ΛCDM era. The equation of state (EOS) parameter (ωd) for DE varies from the phantom (ω < –1) phase to quintessence (ω > –1) phase, which is consistent with observational results. It is found that the discussed model can reproduce the current accelerating phase of the expansion of the universe.

Rotationally symmetric flow of Reiner-Rivlin fluid over a heated porous wall using numerical approach

2021 ◽  
pp. 095440622110342
Author(s):  
Talat Rafiq ◽  
M Mustafa ◽  
Junaid Ahmad Khan
Keyword(s):  
Boundary Layer ◽  
Layer Structure ◽  
Full Range ◽  
Porous Wall ◽  
Symmetric Flow ◽  
Axial Velocity Component ◽  
Suction Velocity ◽  
Rotationally Symmetric ◽  
Non Linear System ◽  
Rotationally Symmetric Flow

This research features one parameter family of solutions representing rotationally symmetric flow of non-Newtonian fluid obeying Reiner-Rivlin model. Such flows take place over a revolving plane permeable surface through origin such that fluid at infinity also undergoes uniform rotation about the vertical axis. Heat transfer accompanied with viscous heating effect is also analyzed. A similarity solution is proposed that results into a coupled non-linear system with four unknowns. Boundary layer structure is characterized by a parameter [Formula: see text] that compares angular velocity of external flow with that of the rotating surface. Solutions are developed by a well-known package bvp4c of MATLAB for full range of [Formula: see text]. Flow pattern for different choices of [Formula: see text] is scrutinized by computing both 2 D and 3 D streamlines. It is further noted that value of suction velocity is important with regards to the sign of axial velocity component. Boundary layer suppresses considerably whenever the surface is permeable. For sufficiently high suction velocity with [Formula: see text], entire fluid undergoes rigid body rotation. In no suction case, axially upward flow accelerates for increasing values of parameter [Formula: see text] in the range [Formula: see text], whereas opposite trend is apparent in the case [Formula: see text]. Results for normalized wall shear and Nusselt number are scrutinized for various choices of embedded parameters.

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