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Determination of Secondary Wavefront Aberrations in Axis-Symmetrical Optical Systems
Issue:
Volume 11, Issue 3, June 2022
Pages:
60-68
Received:
17 April 2022
Accepted:
5 May 2022
Published:
12 May 2022
DOI:
10.11648/j.acm.20221103.11
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Abstract: The design and development of optical systems relies on a thorough theoretical understanding of optical aberrations. However, determining the values of the various-order wavefront aberrations in an optical system is extremely challenging. Accordingly, the present study proposes a methodology for determining the numerical values of the secondary wavefront aberrations of an axis-symmetrical optical system by expanding the optical path length of its general ray using a Taylor series expansion. The determined values of the secondary wavefront aberration coefficients are given. They are distortion W511, field curvature W420, astigmatism W422, coma W331, oblique spherical aberration W240, spherical aberration W060, and six still un-named secondary wavefront aberrations. It is shown that three components (i.e., W244, W153, and W155) are not included among the secondary wavefront aberrations given in the literature despite satisfying the equations of axis-symmetrical nature of axis-symmetrical systems. In other words, the equation of existing literature fails to provide all the components needed to fully compute the secondary wavefront aberrations. By extension, some components of the higher-order wavefront aberrations may also be incompletely presented. The proposed method in this study provides the opportunity to compute all components of various-order wavefront aberrations for rotationally-symmetric optical systems, indicating it is a robust approach for aberration determination.
Abstract: The design and development of optical systems relies on a thorough theoretical understanding of optical aberrations. However, determining the values of the various-order wavefront aberrations in an optical system is extremely challenging. Accordingly, the present study proposes a methodology for determining the numerical values of the secondary wav...
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A Correlation Study of Mathematics and Science Subjects Achievements in Secondary Schools
Vicent Paul Nyakyi,
Amon Mwenda
Issue:
Volume 11, Issue 3, June 2022
Pages:
69-73
Received:
29 March 2022
Accepted:
3 May 2022
Published:
19 May 2022
DOI:
10.11648/j.acm.20221103.12
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Abstract: Mathematics is a subject that is widely applied in solving science problems. There has been an argument that to perform well in sciences, one must also perform well in mathematics. In the light of this assertion, this paper studied the correlation between mathematics and science subjects’ achievements in secondary schools. Two science subjects, physics and chemistry were considered. The study used certificate of secondary education examination results for basic mathematics, physics and chemistry subjects of 229 students from three secondary schools located in Arusha, Tanzania. The schools were chosen based on performance ranks, high performing school, medium performing school and relatively low performing school. The results were for the year 2020. The grades were coded using the scale A – 1, B – 2, C – 3, D – 4, and F – 5. Scatter diagrams and correlation analysis approaches were used to arrive at the conclusion. The study found that there is a moderate positive relationship between mathematics achievements and physics achievements in least performing schools, meaning that students with good performance in mathematics are expected to perform better in physics. On the other hand, the study found a weak positive relationship between mathematics achievements and physics and chemistry achievements in high performing school. The study proposes that performance in mathematics should not be a criterion for selecting students to study science subjects. The results may be useful to educators responsible for selecting students for further studies in science subjects.
Abstract: Mathematics is a subject that is widely applied in solving science problems. There has been an argument that to perform well in sciences, one must also perform well in mathematics. In the light of this assertion, this paper studied the correlation between mathematics and science subjects’ achievements in secondary schools. Two science subjects, phy...
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Analytical and Numerical Solutions for the (3+1)-dimensional Extended Quantum Zakharov-Kuznetsov Equation
Issue:
Volume 11, Issue 3, June 2022
Pages:
74-80
Received:
26 April 2022
Accepted:
18 May 2022
Published:
31 May 2022
DOI:
10.11648/j.acm.20221103.13
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Abstract: The Zakharov-Kuznetsov equation is an important model to describes the nonlinear pulse propagation in plasma physics, which guides the characteristic of weakly nonlinear ion-acoustic waves in plasma composed of cold ions and hot isothermal electrons in a uniform magnetic field. In the current study, we investigate the generalized trigonometric solutions and new travelling wave solutions of the (3+1)-dimensional extended quantum Zakharov- Kuznetsov equation through the (G'/G)-expansion method and the Sech-Tanh expansion method. Before applying these, we imply the traveling wave transformation to convert the (3+1)-dimensional extended quantum Zakharov- Kuznetsov equation to a nonlinear differential equation (NLODE). By the aid of Mathematics software, the dynamical images such as three-dimensional (3D) graphs, two- dimensional (2D) graphs and contour surfaces of local solutions are plotted by choosing the appropriate parameters. The obtained solutions show the simplicity and efficiency of the two approaches that can be applied for nonlinear equations as well as linear ones. Furthermore, the accuracy of the solutions obtained by the two different methods is verified by the Adomain decomposition method (ADM) and showed in tables respectively. The study of ADM method in this paper indivates it is an effective mathematical tool to calculate the numerical solutions and to verify the accuracy of the solutions.
Abstract: The Zakharov-Kuznetsov equation is an important model to describes the nonlinear pulse propagation in plasma physics, which guides the characteristic of weakly nonlinear ion-acoustic waves in plasma composed of cold ions and hot isothermal electrons in a uniform magnetic field. In the current study, we investigate the generalized trigonometric solu...
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Q-borderenergeticity Under the Graph Operation of Complements
Jing Li,
Bo Deng,
Xumei Jin,
Xiaoyun Lv
Issue:
Volume 11, Issue 3, June 2022
Pages:
81-86
Received:
20 May 2022
Accepted:
2 June 2022
Published:
14 June 2022
DOI:
10.11648/j.acm.20221103.14
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Abstract: Let A(G) be the adjacency matrix of graph G. Suppose λn ≤ λn-1 ≤ ··· ≤λ1 are the eigenvalues of A(G). The energy of a graph G is denoted by ε(G), which is defined as the sum of absolute values of its eigenvalues. It is well known that graph energy is found that there are many applications in chemistry. Nikiforov showed that almost all graphs have an energy asymptotically equal to O(n1.5). So, almost all graphs are supperenergetic, i.e., their graph energies are more than those of complete graphs with the same orders. This made an end to the study of supperenergetic graphs. Then the concept of a borderenergetic graph is proposed by Gutman et al. in 2015. If a graph G of order n satisfies it energy ε(G)=2(n-1), then G is called a borderenergetic graph. Recently, Tao and Hou extend this concept to signless Laplacian energy. That is, a graph of order n is called Q-borderenergetic graph if its signless Laplacian energy is equal to that of the complete graph Kn. In this work, by using the graph operation of complements, we find that, for most of Q-borderenergetic graphs, it can not satisfy themselves and their complements are all Q-borderenergetic. Besides, a new lower bound on signless Laplacian energy of the complement of a Q-borderenergetic graph is established.
Abstract: Let A(G) be the adjacency matrix of graph G. Suppose λn ≤ λn-1 ≤ ··· ≤λ1 are the eigenvalues of A(G). The energy of a graph G is denoted by ε(G), which is defined as the sum of absolute values of its eigenvalues. It is well known that graph energy is found that there are many applications in chemistry. Nikiforov showed that almost all graphs have a...
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