On the Schröder–Bernstein property for modules

The well known Schröder–Bernstein Theorem states that any two sets with one to one maps into each other are isomorphic. The question of whether any two (subisomorphic or) direct summand subisomorphic algebraic structures are isomorphic, has long been of interest. Kaplansky asked whether direct summands subisomorphic abelian groups are always isomorphic? The question generated a great deal of interest. The study of this question for the general class of modules has been somewhat limited. We extend the study of this question for modules in this paper. We say that a module Msatisfies the Schröder–Bernstein property (S-B property) if any two direct summands of M which are subisomorphic to direct summands of each other, are isomorphic. We show that a large number of classes of modules satisfy the S-B property. These include the classes of quasi-continuous, directly finite, quasi-discrete and modules with ACC on direct summands. It is also shown that over a Noetherian ring R, every extending module satisfies the S-B property. Among applications, it is proved that the class of rings R for which every R-module satisfies the S-B property is precisely that of pure-semisimple rings. We show that over a commutative domain R, any two quasi-continuous subisomorphic R-modules are isomorphic if and only if R is a PID. We study other conditions related to the S-B property and obtain characterizations of certain classes of rings via those conditions. Examples which delimit and illustrate our results are provided.     

Comparison of Approaches to Quantum Gravitational Corrections on the Reissner-Nordstr?m-de Sitter Black Hole Tunneling Radiation

Although the tunneling approach is fully established for black hole radiation, much work has been done to support the extension of this approach to more general settings. In this letter the Parikh-Kraus-Wilczeck tunneling proposal of black hole tunneling radiation is considered. The Reissner-Nordstr?m-de Sitter black hole thermodynamics is studied according to the generalized uncertainty principle and the modified dispersion relation analysis. It is shown that entropy, temperature and the original Parikh-Kraus-Wilczek calculations of the black hole tunneling probability receive new corrections. The results are compared and it is shown that these two alternative approaches lead to the same results if one uses the suitable expansion coefficients.     

Electron energy spectrum in the field-ionized plasma

In this article, the effect of ionization on the energy spectrum of electrons within the interaction of a laser pulse with hydrogen atoms is investigated using particle-in-cell simulation codes. The results show that the behaviour of electrons' energy distribution function in the field-ionized plasma, which occurred due to the field ionization, compared with that in the pre-plasma strongly depends on the pulse shape. For short rise-time pulses (here 30 fs), due to the rapid enhancement of laser electric field, ionization occurs quickly, and as a result, there is not much difference in the electron energy in both the media. However, for pulses with rise time of 40 fs, in the pre-plasma state, the electron population reaches higher energies compared with the field-ionized plasma state. The main reason for this difference is the nonlinear wave breaking that happens earlier due to density inhomogeneity in the field-ionized plasma. On the other hand, at longer rise-time pulses (here 60 and 70 fs), electrons achieve higher energies in the field-ionized plasma than those in the case of pre-plasma. In this case, because of density fluctuations in the field-ionized plasma, the Raman backscattered radiations are seeded by a strong initial noise at the earlier times and the Mendonca condition for chaos threshold is met sooner. Therefore, the electrons gain more energy through the stochastic mechanism that is in agreement with chaotic nature of the motion.     

Damping in the Interaction of a Two-Photon Field and a Two-Level Atom Through Quantized Caldirola-Kanai Hamiltonian

Following the lines of the recent papers (Daneshmand and Tavassoly, Int. J. Theor. Phys. 56, 1218 (2017)), we study quantum mechanical treatments of an interaction between a two-level atom with a single-mode field in the two-photon Jaynes-Cummings model, where the Hamiltonian of the field is considered to be the quantized Caldirola-Kanai (CK) Hamiltonian. As a result, we would expect that the quantum dynamics of the two-photon JCM in terms of the CK Hamiltonian is qualitatively different from that of the usual one-photon case. We analytically calculate the explicit form of the atom-field entangled state and numerically evaluate the dynamics of its physical properties. The degree of entanglement, atomic population as well as sub-Poissonian statistics and quadrature squeezing of the field are analyzed. We adjust the latter evolved parameters by appropriately tuning the damping parameter within the CK Hamiltonian and detuning factor. Finally, we report a field detuning asymmetry in the collective statistical behavior.

     

Synthesis of intermediate sitting-atop complexes (i-SAT) from the reaction between free base meso-tetraarylporphyrins and phosphorus(III) chloride in solvent free media

The reaction of meso-tetraarylporphyrins (H₂tap) with phosphorus(III) chloride has been studied in the solid state. The reaction products are intermediate sitting-atop (i-SAT) complexes in which two pyrrolenine nitrogen atoms of the porphyrins act as electron donors to the phosphorus(III) atoms and two protons on the pyrrole nitrogen atoms remain. These new complexes have been characterized by (¹H, ³¹P, ¹³C) NMR, FT-IR and UV–Vis spectroscopy, elemental analysis and electrical conductometry.     

Solid phase extraction of copper (II) by sorption on octadecyl silica membrane disk modified with a new Schiff base and determination with atomic absorption spectrometry

A simple, selective and reliable method for rapid extraction and determination of trace amounts of Cu (II) ions from aqueous samples using octadecyl-bonded silica membrane disks modified with bis-(3-methoxy salicylaldehyde)-1,6-diaminohexane and flame atomic absorption spectrometry (FAAS) is presented. Extraction efficiency, the influence of pH, flow rates, amount of ligand, and type and least amount of eluant were investigated. The linear dynamic range of the proposed method for Cu (II) ions was found in a wide concentration range of 1.0 (± 0.2)–150 (± 2) μg l^− 1. The detection limit and preconcentration factor of this method were found 30.0 (± 0.7) ng l^− 1 and 100 respectively. The reproducibility of the procedure is at the most 2.0%. The effects of various cationic interferences on the percent recovery of copper ion were studied. The method was used to the recovery of copper ion from different synthetic, alloys and biological samples.     

Infrared spectra of the OCS trimer

Infrared spectra of the barrel-shaped trimer (OCS)(3), previously known from its microwave spectrum, are reported for the first time. The observations are carried out in a supersonic slit-jet expansion of a He+OCS gas mixture which is probed with a tunable diode laser. Three rotationally resolved bands associated with the nu(1) fundamental vibration of OCS (2062.20 cm(-1)) are observed, at about 2047, 2053, and 2077 cm(-1). Small perturbations are noted in the 2077 cm(-1) band and may also be present in the 2053 cm(-1) band, which is weak and hence more difficult to analyze precisely. Employing a variety of evidence, we suggest a plausible assignment for the nature of the OCS vibrations in each of the three bands.     

Comments on “Barut-Girardello Coherent States for the Parabolic Cylinder Functions”

In Chenaghlou and Faizy (Int. J. Theor. Phys. 2008), the authors claim that they have constructed the Barut-Girardello coherent states for the parabolic cylinder functions. However, we point out here that by introducing these coherent states, Schrödinger was able to put forth the idea of “coherent states of the quantum harmonic oscillator” over eighty years ago. These coherent states are derived not only from the Barut-Girardello eigenvalue equation, but also from the Schrödinger and the Klauder-Perelomov approaches. Thus, contrary to their claim, the authors have not introduced new coherent states. In particular, a wide range of the parabolic cylinder functions do not form an orthonormal basis.     

Horizon thermodynamics and gravitational field equations in quasi-topological gravity

In this paper we show that the gravitational field equations of $(n+1)$ -dimensional topological black holes with constant horizon curvature, in cubic and quartic quasi-topological gravity, can be recast in the form of the first law of thermodynamics, $dE=TdS-PdV$ , at the black hole horizon. This procedure leads to extract an expression for the horizon entropy as well as the energy (mass) in terms of the horizon radius, which coincide exactly with those obtained in quasi-topological gravity by solving the field equations and using the Wald’s method. We also argue that this approach is powerful enough to be extended to all higher order quasi-topological gravity for extracting the corresponding entropy and energy in terms of horizon radius.     

Kinked crack analysis by a hybridized boundary element/boundary collocation method

In this research a two dimensional displacement discontinuity method (which is a kind of indirect boundary element method) using higher order elements (i.e. a source element with a cubic variation of displacement discontinuities having four sub-elements) is used to obtain the displacement discontinuities along each boundary element. In this paper, three kinds of the higher order boundary elements are used: the ordinary elements, the kink elements and the special crack tip elements.The boundary collocation technique is used for the calculation of the displacement discontinuities at the center of each sub-elements. Again a special boundary collocation technique is used to treat the kinked source elements occur in the crack analysis. Considering the two source elements (each having four sub-elements) joined at a corner (kink point). The collocation points in the cubic element model which are outside of the kink point are moved to the crack kink then the displacement discontinuities on the left and right sides of the kink are calculated. The displacement discontinuities of the kink point are obtained by averaging the corresponding values of its left and right sides. The special crack tip elements are also treated by the boundary displacement collocation technique considering the singularity variation of the displacements and stresses near the crack tip. Some simple example problems are solved numerically by the proposed method. The numerical results are compared with the corresponding results obtained by the previous methods cited in the literature. This comparison shows a very good agreement between the results and verify the accuracy and validity of the proposed method.