S-states of helium-like ions

A simple Mathematica (version 7) code for computing S-state energies and wave functions of two-electron (helium-like) ions is presented. The elegant technique derived from the classical papers of Pekeris (1958, 1959, 1962, 1965, 1971) [1], [2] and [3] is applied. The basis functions are composed of the Laguerre functions. The method is based on the perimetric coordinates and specific properties of the Laguerre polynomials. Direct solution of the generalized eigenvalues and eigenvectors problem is used, distinct from the Pekeris works. No special subroutines were used, only built-in objects supported by Mathematica. The accuracy of the results and computation times depend on the basis size. The ground state and the lowest triplet state energies can be computed with a precision of 12 and 14 significant figures, respectively. The accuracy of the higher excited states calculations is slightly worse. The resultant wave functions have a simple analytical form, that enables calculation of expectation values for arbitrary physical operators without any difficulties. Only three natural parameters are required in the input.The above Mathematica code is simpler than the earlier version (Liverts and Barnea, 2010 [4]). At the same time, it is faster and more accurate.

Program summary

Program title: TwoElAtomSL(SH)Catalogue identifier: AEHY_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEHY_v1_0.htmlProgram obtainable from: CPC Program Library, Queen?s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 11 434No. of bytes in distributed program, including test data, etc.: 540 063Distribution format: tar.gzProgramming language: Mathematica 7.0Computer: Any PCOperating system: Any which supports Mathematica; tested under Microsoft Windows XP and Linux SUSE 11.0RAM:?¹⁰⁹ bytesClassification: 2.1, 2.2, 2.7, 2.9Nature of problem: The Schrödinger equation for atoms (ions) with more than one electron has not been solved analytically. Approximate methods must be applied in order to obtain the wave functions or another physical attributes from quantum mechanical calculations.Solution method: The S-wave function is expanded into a triple set of basis functions which are composed of the exponentials combined with the Laguerre polynomials in the perimetric coordinates. Using specific properties of the Laguerre polynomials, solution of the two-electron Schrödinger equation reduces to solving the generalized eigenvalues and eigenvector problem for the proper Hamiltonian. The unknown exponential parameter is determined by means of minimization of the corresponding eigenvalue (energy).Restrictions: First, the too large length of expansion (basis size) takes the too large computation time and operative memory giving no perceptible improvement in accuracy. Second, the number of shells Ω in the wave function expansion enables one to calculate the excited nS-states up to n=Ω+1 inclusive.Running time: 2–60 minutes (depends on basis size and computer speed).     

Efficient Solution of A, x ( k )?= b ( k ) Using A ?1

In this work, we consider the problem of solving , , where b ^(k+1) = f(x ^(k)). We show that when A is a full matrix and , where depends on the specific software and hardware setup, it is faster to solve for by explicitly evaluating the inverse matrix A ⁻¹ rather than through the LU decomposition of A. We also show that the forward error is comparable in both methods, regardless of the condition number of A.     

Renaming and the weakest family of failure detectors

We address the question of the weakest failure detector to circumvent the impossibility of $(2n-2)$ -renaming in a system of up to $n$ participating processes. We derive that in a restricted class of eventual failure detectors there does not exist a single weakest oracle, but a weakest family of oracles $\zeta _n$ : every two oracles in $\zeta _n$ are incomparable, and every oracle that allows for solving renaming provides at least as much information about failures as one of the oracles in $\zeta _n$ . As a by product, we obtain one more evidence that renaming is strictly easier to solve than set agreement.     

Software transactional memory

Summary.  As we learn from the literature, flexibility in choosing synchronization operations greatly simplifies the task of designing highly concurrent programs. Unfortunately, existing hardware is inflexible and is at best on the level of a Load–Linked/Store–Conditional operation on a single word. Building on the hardware based transactional synchronization methodology of Herlihy and Moss, we offer software transactional memory (STM), a novel software method for supporting flexible transactional programming of synchronization operations. STM is non-blocking, and can be implemented on existing machines using only a Load–Linked/Store–Conditional operation. We use STM to provide a general highly concurrent method for translating sequential object implementations to non-blocking ones based on implementing a k-word compare&swap STM-transaction. Empirical evidence collected on simulated multiprocessor architectures shows that our method always outperforms the non-blocking translation methods in the style of Barnes, and outperforms Herlihy’s translation method for sufficiently large numbers of processors. The key to the efficiency of our software-transactional approach is that unlike Barnes style methods, it is not based on a costly “recursive helping” policy. Received: January 1996 / Revised: June 1996 / Accepted: August 1996     

Compressive system identification: Sequential methods and entropy bounds

In the first part of this work, a novel Kalman filtering-based method is introduced for estimating the coefficients of sparse, or more broadly, compressible autoregressive models using fewer observations than normally required. By virtue of its (unscented) Kalman filter mechanism, the derived method essentially addresses the main difficulties attributed to the underlying estimation problem. In particular, it facilitates sequential processing of observations and is shown to attain a good recovery performance, particularly under substantial deviations from ideal conditions, those which are assumed to hold true by the theory of compressive sensing. In the remaining part of this paper we derive a few information-theoretic bounds pertaining to the problem at hand. The obtained bounds establish the relation between the complexity of the autoregressive process and the attainable estimation accuracy through the use of a novel measure of complexity. This measure is used in this work as a substitute to the generally incomputable restricted isometric property.     

Approximate analytical solutions to the condensation-coagulation equation of aerosols

We present analytical solutions to the steady state nucleation-condensation-coagulation equation of aerosols in the atmosphere. These solutions are appropriate under different limits but more general than previously derived analytical solutions. For example, we provide an analytic solution to the coagulation limit plus a condensation correction. Our solutions are then compared with numerical results. We show that the solutions can be used to estimate the sensitivity of the cloud condensation nuclei number density to the nucleation rate of small condensation nuclei and to changes in the formation rate of sulfuric acid.

© 2016 American Association for Aerosol Research     

Effect of oxidants on microalgal flocculation

The effects of chlorine, ozone and chlorine dioxide on Scenedesmus sp. cultures were studied. Algal cell viability and chlorophyll concentration decreased, and the concentration of dissolved organic substances increased with increasing applied oxidant concentration. Pretreatment with chlorine dioxide (1, 3 or 5 mg l⁻¹) or ozone (2.6, 4.6 or 8.1 mg l⁻¹) on algal cultures enhanced algal flocculation with alum, while prechlorination with 10 or 20 mg l⁻¹ increased the required dosage of alum by 15%. Scanning electron micrographs of oxidized cells revealed drastically adverse effects upon the cell surface architecture: in addition to the oxidation of noncellular organic materials, the oxidants damaged both cell surface structures and intracellular components. A model explaining the effects of the different oxidants on microalgal flocculation is suggested.