898 research outputs found

    A Study on Set-Graphs

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    A \textit{primitive hole} of a graph GG is a cycle of length 33 in GG. The number of primitive holes in a given graph GG is called the primitive hole number of that graph GG. The primitive degree of a vertex vv of a given graph GG is the number of primitive holes incident on the vertex vv. In this paper, we introduce the notion of set-graphs and study the properties and characteristics of set-graphs. We also check the primitive hole number and primitive degree of set-graphs. Interesting introductory results on the nature of order of set-graphs, degree of the vertices corresponding to subsets of equal cardinality, the number of largest complete subgraphs in a set-graph etc. are discussed in this study. A recursive formula to determine the primitive hole number of a set-graph is also derived in this paper.Comment: 11 pages, 1 figure, submitte

    Wavefront errors in a two-beam interferometer

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    This paper deals with the impact of wavefront errors, due to the optical aberrations of a two-beam interferometer, on the period of the travelling fringe observed by integrating the interference pattern. A Monte Carlo simulation of the interferometer operation showed that the fringe-period estimate is unbiased if evaluated on the basis of the angular spectrum of the beam entering the interferometer, but the wavefront errors increase the uncertainty

    Forward scattering in two-beam laser interferometry

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    A fractional error as large as 25 pm mm(-1) at the zero optical-path difference has been observed in an optical interferometer measuring the displacement of an x-ray interferometer used to determine the lattice parameter of silicon. Detailed investigations have brought to light that the error was caused by light forward-scattered from the beam feeding the interferometer. This paper reports on the impact of forward-scattered light on the accuracy of two-beam optical interferometry applied to length metrology, and supplies a model capable of explaining the observed error

    Density functional theory calculations of the stress of oxidised (1 1 0) silicon surfaces

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    The measurement of the lattice-parameter of silicon by x-ray interferometry assumes the use of strain-free crystals. This might not be the case because surface relaxation, reconstruction, and oxidation cause strains without the application of any external force. In a previous work, this intrinsic strain was estimated by a finite element analysis, where the surface stress was modeled by an elastic membrane having a 1 N m(-1) tensile strength. The present paper quantifies the surface stress by a density functional theory calculation. We found a value exceeding the nominal value used, which potentially affects the measurement accuracy
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