On Fuzzy Soft Linear Operators in Fuzzy Soft Hilbert Spaces

Due to the difficulty of representing problem parameters fuzziness using the soft set theory, the fuzzy soft set is regarded to be more general and flexible than using the soft set. In this paper, we define the fuzzy soft linear operator T~ in the fuzzy soft Hilbert space H~ based on the definition of the fuzzy soft inner product space U~,·,·~ in terms of the fuzzy soft vector v~fGe modified in our work. Moreover, it is shown that ℂnA, ℝnA and ℓ2A are suitable examples of fuzzy soft Hilbert spaces and also some related examples, properties and results of fuzzy soft linear operators are introduced with proofs. In addition, we present the definition of the fuzzy soft orthogonal family and the fuzzy soft orthonormal family and introduce examples satisfying them. Furthermore, the fuzzy soft resolvent set, the fuzzy soft spectral radius, the fuzzy soft spectrum with its different types of fuzzy soft linear operators and the relations between those types are introduced. Moreover, the fuzzy soft right shift operator and the fuzzy soft left shift operator are defined with an example of each type on ℓ2A. In addition, it is proved, on ℓ2A, that the fuzzy soft point spectrum of fuzzy soft right shift operator has no fuzzy soft eigenvalues, the fuzzy soft residual spectrum of fuzzy soft right shift operator is equal to the fuzzy soft comparison spectrum of it and the fuzzy soft point spectrum of fuzzy soft left shift operator is the fuzzy soft open disk λ~<~1~. Finally, it is shown that the fuzzy soft Hilbert space is fuzzy soft self-dual in this generalized setting.

BD-21.3873: An Heavy Element-Rich Symbiotic?

The recent discovery (Jorissen and Mayor, 1988) that not only Ba II but also S stars appear to belong predominantly to binary systems with periods of several hundred days raises the question of the similarity between these peculiar red giants and symbiotic stars. That question has been addressed by looking whether symbiotics have enhanced s-element lines in their spectra. In most cases, definite conclusions are hampered by the composite nature of such spectra. Nevertheless, in the case of the “yellow symbiotic” BD -21.3873, the spectral classification criteria provided by Keenan and Wilson (1977) allow to assign without ambiguity the type GIII-II to BD -21., 3873, the symbiotic nature of that star being apparent only through weak Fe II emission lines. Some heavy element lines (such as Sm II λ4220.7, λ4221.1 and λ4566.3) are clearly enhanced. Comparison with a spectrum of a G3Ib supergiant ensures that luminosity is not responsible for that effect. Moreover, some of these spectral features are also enhanced in Barium stars, as seen on a comparison spectrum of a Barium Star.

The Calibration of Stellar Spectra Used in Measurements of the Generalized Compton Shift

In the study of the changes induced by the Compton effect and Thomson scattering on the shape of spectral lines in light traversing a chromosphere or a planetary nebula it is necessary to have accurate wavelength measurements, central intensities and half widths (FWHM) of the lines. In the comparison of FWHM measures belonging to different spectra it is useful to have the intensities of the continuum at the wavelengths of the lines. The studied spectra should have a dispersion higher than 10 Å/mm and a spectral range larger than 700 Å. A short recommendation is also given about the comparison spectrum and the calibration plate.

A Determination of the Radial Velocities of Galaxies

Fourteen galaxies were studied using 24 spectra, obtained with the 6 meter telescope of the Special Astrophysical Observatory of the USSR Academy of Sciences. The dispersion of the spectrograph in the primary focus of the telescope was 189 A/mm.The neon spectrum was studied in order to determine the wavelengths of the comparison spectrum and to plot a dispersion curve. The following Ne I lines were used for a comparison: 6143, 6266, 6506, 6678, 6717, 6929 and 7032 A. In galactic spectra (wavelength range of 4300 - 7000 A) 21 absorption lines were detected. They belong to H, Na I, Ca I, Ca II, Mg I and otner chemical elements. Three emission lines of the night sky with wavelengths of 5577, 5890 and 6300 A are seen in the spectra of galaxies. They were also used as reference lines for our determination.

Quantitative Two-Dimensional Classifications of Low Dispersion Objective Prism Spectra

The objective prism (or grating) is the most efficient stellar spectrograph since it suffers no light losses at a slit and - compared to a slit spectrograph - only small losses in the camera optics. However, objective prism spectra have two principal disadvantages:(1)the spectral resolution (and consequently the sharpness of spectral features) depends on the seeing, which may vary from plate to plate.(2)quantitative measurements are difficult because of the difficulty in obtaining an exact photographic calibration; no comparison spectrum can be placed near the stellar spectra.

On temperature and pressure regulation in prismatic spectrographs

It is well known that displacements of spectrum lines during a photographic exposure can be caused by changes in the temperature of the dispersing apparatus and of the air in the spectrograph. In the case of grating instruments elaborate precautions are sometimes taken to keep the room in which the spectrograph is housed at a constant temperature. Birge* has described the precautions taken at the University of Wisconsin, where the grating spectrograph is mounted in a double-walled room, the space between the walls being kept constant by suitable electric heating. In the photography of faint spectra prismatic spectrographs are sometimes to be preferred to gratings on account of their superior light-gathering power, and it is of importance to consider the errors in measurement and loss of definition due to variations in temperature. The change in refractive index with temperature appears to vary rather considerably with different melts of glass. Giffordf has measured the temperature coefficients for a series of glasses in air, and his results show that these are positive for all the glasses investigated, with one exception, Fluor Crown, which would not be suitable for the construction of prisms for spectrographs. The change in refractive index per degree Centigrade for glasses of the type suitable for the construction of prisms is shown to be of the order of 5 X 10 -6 , and since for a typical glass at λ = 5875 A, the change per Ångstrom unit is 9 X 10 -6 , the shift in wavelength per degree Centigrade would be upwards of half an Ångström unit. The temperature coefficient of some glasses seems to be considerably greater than the above value. Displacements of spectrum lines are also effected by changes in the barometric pressure during long exposures. There are, of course, many days on which this effect may be neglected, but this source of error does not appear to have received the attention that it merits. The refractive index of air at 760 mm. pressure and 20° C. is about 1•000275, so that if we assume that n p = n 760 — [( n — 1) (760 — p)/760] the change in n per cm. pressure is about 3·6 X 10 -6 , and taking the change in n per Ångström unit at D 3 as 9 X 10 -6 , we expect a shift of about 0·4 A per cm. pressure. A change in pressure as great as 1 cm. during the course of an exposure would be unusual,but changes exceeding 1 mm. are not infrequent during long exposures, and it is evident that they cannot be ignored. It is usual to expose the comparison spectrum, comprising the lines of standard wavelength, either simultaneously with the spectrum under investigation, or in instalments at regular intervals, but it has been shown that errors due to displacements* can only be eliminated in this way in the case of lines which are symmetrical, and in any case there is a loss of definition and resolving power with a loss of accuracy of the settings in the course of measurement. The experiments described in this communication relate to a prism spectrograph recently constructed for the writer by Messrs. Adam Hilger, and although the methods adopted and the conclusions reached seem fairly obvious, they may be worthy of record in view of the fact that satisfactory results were obtained only after trials extending over many months, and the details of construction appear to be of vital importance.

On errors arising in the measurement unsymmetrical spectrum lines

It is generally recognised that one of the most serious sources of error in relative determinations of the wave-lengths of spectrum lines by micrometric measurements of the positions of the lines on photographic plates lies in the mechanical and other displacements, which result in errors in the superposition or juxtaposition of the spectrum under investigation and the standard spectrum, the wave-lengths of which are supposed to be known. Thus, if a spectrum to be measured is photographed on a plate and an iron arc spectrum is then photographed in juxtaposition, an error may arise, either from some flexure or mechanical displacement between the two exposures or from a gradual change of temperature of the dispersing prism or grating, resulting in a continuous drift of the spectrum across the plate. Even when the utmost care is taken to avoid these displacements by rigid design of the spectrograph and proper control of temperature, it is hardly ever possible to eliminate them entirely, and it is necessary to follow some procedure which will reduce the errors arising from them to a minimum. A method which is commonly used is to expose the comparison spectrum in two parts. For example, a comparison spectrum with one-half the total exposure required may be taken first; the spectrum to be measured is then exposed for an appropriate time, and finally the second half-exposure of the comparison spectrum is superposed on the first. Images of the comparison lines (and in the case of a long exposure of the lines to be measured) would be somewhat broadened, but it might at first seem that the maxima of the lines in the two spectra would be in correct juxtaposition, supposing that there has been a constant drift of the spectrum across the plate owing to a gradual change of temperature. St. John and Babcock, in a study of the pole effect in the iron arc, used a system of reflecting prisms to project on theslit simultaneously the light from the neighbourhood of the pole in juxtaposition with that from the centre of the arc, the intensities being equalised by means of a rotating sector, which occulted for an appropriate period the more intense light from the neighbourhood of the pole.