Approximate Cross Research Articles

Mechanism behind the Beauty: The Golden Ratio Appeared in the Shape of Red Blood Cells

AbstractIn the past two decades, a rigorous solution for the shape of human red blood cell (RBC) with a negative spontaneous curvature c0 has been derived with the Helfrich model under the condition that both the osmotic pressure Δp and tensile stress λ are equal to zero. By fitting the experimentally observed shape of RBC, c0R0 has been predicted to be –1.62, theminus golden ratio, where R0 is the radius of a sphere which has the same surface area as RBC. In this paper, we verify this prediction by comparing experimental data with an analytical equation describing the relation between volume and surface area. Furthermore, it is also found ρmax /ρB ≈ 1.6 with ρmax the maximal radius and ρB the characteristic radius of RBC, showing an approximate beautiful golden cross section of RBC. On the basis of a complete numerical calculation, we propose a mechanism behind the beauty of the minus golden ratio that c0R0 results from the balance between the minimization of the surface area and the requirement of adequate deformability of RBC to allow it passing through the spleen, the so called “physical fitness test”.

Time-dependent close-coupling calculations of positronium formation in positron-hydrogen collisions

Positronium-formation cross sections from positron-hydrogen collisions are calculated by solving a set of close-coupled, time-dependent partial differential equations. These cross sections are determined by propagating a time-evolving wave packet on a two-dimensional radial lattice and then projecting the wave function onto the stationary states of the atom. Calculations are performed through $L=8$ and are extrapolated for $Lg~9$ at energies of 30, 40, and 50 eV. Using distorted-wave calculations to determine ionization cross sections at these higher energies, we are able to extract approximate positronium-formation cross sections from the transfer ionization cross sections. Comparisons are made with experiment.

Approximate multijet cross sections in QCD.

A method is given to approximate the matrix elements squared for all parton processes involving a quark-antiquark pair plus an arbitrary number of gluons. Detailed comparisons are made between the results of this method and the exact results which are known for the quark-antiquark plus-four-gluon-processes in four-jet production at hadron colliders. Together with Maxwell's approximate result for six-gluon processes an excellent agreement is found for the total cross section and shape of four-jet production at hadron colliders.

Fuzzy Tree Automata and Syntactic Pattern Recognition

An approach of representing patterns by trees and processing these trees by fuzzy tree automata is described. Fuzzy tree automata are defined and investigated. The results include that the class of fuzzy root-to-frontier recognizable ¿-trees is closed under intersection, union, and complementation. Thus, the class of fuzzy root-to-frontier recognizable ¿-trees forms a Boolean algebra. Fuzzy tree automata are applied to processing fuzzy tree representation of patterns based on syntactic pattern recognition. The grade of acceptance is defined and investigated. Quantitative measures of ``approximate isosceles triangle,'' ``approximate elongated isosceles triangle,'' ``approximate rectangle,'' and ``approximate cross'' are defined and used in the illustrative examples of this approach. By using these quantitative measures, a house, a house with high roof, and a church are also presented as illustrative examples. In addition, three fuzzy tree automata are constructed which have the capability of processing the fuzzy tree representations of ``fuzzy houses,'' ``houses with high roofs,'' and ``fuzzy churches,'' respectively. The results may have useful applications in pattern recognition, image processing, artificial intelligence, pattern database design and processing, image science, and pictorial information systems.

A Low-Order Approximation for Highly Anisotropic Multigroup Transport Cross Sections

It is shown that for transport problems involving fine-energy-group structure or light element scattering, the elastic scattering transfer cross sections can be approximated by trapezoid-shaped distributions. In particular, for group structures of equal lethargy widths, the trapezoidal distribution reduces to a triangular shape. These approximate transfer cross sections are readily incorporated into a discrete ordinates code and often produce results superior to those obtained with conventional Legendre expansion techniques.

Measurements of relative subshell photoionization cross sections in the noble gases by ultraviolet photoelectron spectroscopy

A method is described for the measurement of relative subshell photoionization cross sections (SPCS) in the noble gases by the technique of ultraviolet photoelectron spectroscopy. Values are reported for the photoionization cross sections of the neon $2p$, krypton $4p$, xenon $5p$, and xenon $5s$ levels relative to that of the argon $3p$ level. The photon energies used for these measurements were 21.22, 40.81, and 48.37 eV. Approximate absolute subshell photoionization cross sections deduced from these relative cross sections are compared to other absolute SPCS obtained from total photoabsorption measurements and also from theory. Results of some relative photoionization cross-section measurements for the outer $p$ bands associated with the halide ions in solid alkali halides and the outer $d$ bands of metallic gold and silver are also presented. Close similarities in variation with photon energy and atomic number are found between the relative photoionization cross sections of the outer $p$ bands of the alkali halides and the outer $p$ levels of the noble gases, but the photoionization cross sections of the $d$ bands of both gold and silver exhibit quite different properties from those found for the outer $p$ bands and levels of both alkali halides and noble gases. A general conclusion is drawn concerning cross sections for photoionization from electron energy bands in solids.

Ba+Resonance-Line Excitation in Plasmas

The rate of excitation by electron collisions of the $6^{2}P$ levels of the ${\mathrm{Ba}}^{+}$ ion is experimentally determined as a function of electron temperature in the range 2500-6000\ifmmode^\circ\else\textdegree\fi{} K. The measured rate is the sum of the excitation rates from the $6^{2}S$ ground state and from the metastable $5^{2}D$ states. The rate coefficients, as well as the corresponding electron temperatures, are obtained from population densities of the $6^{2}P$, $6^{2}S$, and $5^{2}D$ levels. The $6^{2}S$ and $5^{2}D$ population densities are determined from resonance fluorescence measurements and the $6^{2}P$ densities from the emitted light intensities. Approximate near-threshold excitation cross sections are deduced from the rate coefficients. Either the absolute rate coefficients or the ratio of excitation rates to the two $6^{2}P$ levels is suitable for electron thermometry in low-density plasmas where photoexcitation and radiation trapping are negligible.

Born Cross-Sections for Selected Inelastic Collisions of Protons with Hydrogen Atoms in States up to N = 10.

Born total cross sections for inelastic collisions of protons with excited hydrogen atoms may be derived approximately from the Born cross sections for inelastic collisions of electrons with excited hydrogen atoms. A comparison of the exact and approximate Born cross sections for the collisions p + H(nl) yields p + H(n'l'): 1s--2s, 1s--2p, 3s--4p, 3d--4f, shows that they agree down to energies near the Born maximum. By using the known electron cross sections, approximate Born total cross sections are calculated for proton collisions for some of the transitions 1--n'; 2--n'; 3--4, 5; 4--5, 8; 5--8; 10-- 11. In addition, a simple formula is given for finding other approximate proton cross sections in terms of the approximate electron cross sections calculated recently. (auth)

10-Mev Proton Reaction Cross Sections Compared with Surface and Volume Absorption Optical Models of the Nucleus

The (p,n) cross sections were measured at 9.85 Mev for self-supporting thin targets of Al, Ti, Fe, Co, Ni, Cu/sup 63/, Cu/sup 65/, Rh, Ag, Sn, Ta, and Au. The (p,2n) contributions were calculated using the statistical model of the nucleus for Rh, Ta, Ag, and Au. Charged-particle emission was assumed negligible in Ta and Au because of small Coulomb penetrabilities. Approximate proton reaction cross sections were obtained by adding (p,n) and (p,2n) cross sections to (p,p') and (p, alpha ) cross sections previously reported by Meyer and Hintz. These results were compared with volume absorption and surface absorption optical- model calculations of proton reaction cross sections. The parameters for both model calculations were obtained prior to this work by fitting proton elastic scattering and polarization data. The results indicate a surface-absorption potential rather than a volume-absorption potential. (auth)

The composition of the Orion Nebula.

view Abstract Citations References Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The composition of the Orion Nebula. Aller, Lawrence H. ; Liller, William Abstract Photoelectric spectrophotometric measurements of line intensities in the Orion Nebula are combined with Wyse's photographic measurements for the weaker lines to obtain the concentrations of various ions of the permanent gases. As noted by Osterbrock and others, there is evidence for large density fluctuations in this object. We have adopted a mean electron density NE = 4 X i~~ from the forbidden lines and TE = 90000K from the photoelectrically measured ~4363/ (N1 + N2) ratio. Approximate cross- sections have had to be used for sulfur, chlorine, and argon. A fairly well-defined empirical ionization curve was obtained. The following table compares the nebular abundances with those of early type stars as determined in the Michigan program: logN logN Star Nebula Star Nebula Nitrogen 9.2 9.1 Sulphur 9.0 9.1 Oxygen 10.0 10.0 Chlorine 8.7 7.3 Neon 10.1 10.2 Argon 8.4 8.6 University of Michigan Observatory, Ann Arbor, Mich. Publication: The Astronomical Journal Pub Date: 1959 DOI: 10.1086/108011 Bibcode: 1959AJ.....64R..49A full text sources ADS |