A novel algorithm for solving quasi penta-diagonal linear systems

In this paper, a novel numerical algorithm for solving quasi penta-diagonal linear systems is presented. The computational costs of the algorithm is less than those of three successful algorithms given by El-Mikkawy and Rahmo (Comput Math Appl 59:1386–1396, 2010), by Lv and Le (Appl Math Comput 204:707–712, 2008), and by Jia et al. (Int J Comput Math 89:851–860, 2012). In addition, a new recursive method for inverting the quasi penta-diagonal matrices is also discussed. The implementation of the algorithm using Computer Algebra Systems (CASs) such as MATLAB and MAPLE is straightforward. Two numerical examples are given in order to demonstrate the performance and efficiency of our algorithm.     

A symbolic algorithm for periodic tridiagonal systems of equations

In the current paper, we present a novel symbolic algorithm for solving periodic tridiagonal linear systems without imposing any restrictive conditions. The computational cost of the algorithm is less than or almost equal to those of three well-known algorithms given by Chawla and Khazal (Int. J. Comput. Math. 79(4):473–484, 2002) and by El-Mikkawy (Appl. Math. Comput. 161:691–696, 2005), respectively. In addition, the solution of periodic anti-tridiagonal linear systems is also discussed. Two numerical experiments are provided in order to illustrate the performance and effectiveness of our algorithm. All of the experiments were performed on a computer with aid of programs written in MATLAB.     

Comment on “A note on the principal measure and distributional (p,q)-chaos of a coupled lattice system related with Belusov–Zhabotinskii reaction”

In García Guirao and Lampart (J Math Chem 48:159–164, 2010) presented a lattice dynamical system stated by Kaneko (Phys Rev Lett 65:1391–1394, 1990) which is related to the Belusov–Zhabotinskii reaction. In this note, we give an example which shows that the proofs of Theorems 3.1 and 3.2 in [J Math Chem 51:1410–1417, 2013] are incorrect, and two open problems.     

Li–Yorke chaos in a coupled lattice system related with Belusov–Zhabotinskii reaction

Garca Guirao and Lampart (J Math Chem 48:66–71, 2010; J Math Chem 48:159–164, 2010) said that for non-zero couplings constant, the lattice dynamical system is more complicated. Motivated by this, in this paper, we prove that this coupled map lattice system is Li–Yorke chaotic for coupling constant ${0 < \epsilon <1 }$ .     

A strategy for selecting the frequency in trigonometrically-fitted methods based on the minimization of the local truncation errors and the total energy error

Trigonometrically-fitted methods have been largely used for solving second-order differential problems, and particularly for solving the radial Schrödinger equation (see for instance Alolyan and Simos in J Math Chem 50:782–804, 2012; Simos in J Math Chem 34:39–58, 2003, 44:447–466, 2008; Vigo-Aguiar and Simos in J Math Chem 29:177–189, 2001, 32:257–270, 2002 and the references therein contained). It is well-known that for periodic or oscillatory problems, trigonometrically fitted methods are more efficient than non-fitted methods. A large number of different approaches have been considered in the scientific literature to obtain analytical approximations to the frequency of oscillation in case of periodic solutions, which are valid for a large range of amplitudes of oscillation. However, these techniques have been limited to obtaining only one or two iterates because of the great amount of algebra involved. In this paper we consider the use of a trigonometrically fitted method to obtain numerical approximations for the solutions. This yields very acceptable results provided that the approximation of the parameter of the method is done with great accuracy. Many trigonometrically fitted methods have been reported in the literature, but there is no decisive way to obtain the optimal frequency value. We present a strategy for the choice of the parameter value in the adapted method, based on the minimization of the sum of the total energy error and the local truncation errors in the solution and in the derivative. We include an example solved numerically that confirms the good performance of the strategy adopted.     

The proof of a conjecture on the comparison of the energies of trees

The energy of a graph is defined as the sum of the absolute values of the eigenvalues of the graph. In this paper, we first present a new method to directly compare the energies of two bipartite graphs, then also present some new techniques to compare the quasi-orders of some bipartite graphs. As the applications of these methods, we prove that a conjecture proposed by Wang and Kang (J Math Chem 47(3):937–958, 2010) is true. At the same time, our results also provide the simplified proofs of the main results of Wang and Kang (J Math Chem 47(3):937–958, 2010) and Li and Li (Electron J Linear Algebra 17:414–425, 2008).     

An approximation of the Cioslowski–Mixon bond order indexes using the AlteQ approach

Fast and reliable prediction of bond orders in organic systems based upon experimentally measured quantities can be performed using electron density features at bond critical points (J Am Chem Soc 105:5061–5068, 1983; J Phys Org Chem 16:133–141, 2003; Acta Cryst B 61:418–428, 2005; Acta Cryst B 63:142–150, 2007). These features are outcomes of low-temperature high-resolution X-ray diffraction experiments. However, a time-consuming procedure of gaining these quantities makes the prediction limited. In the present work we have employed an empirical approach AlteQ (J Comput Aided Mol Des 22:489–505, 2008) for evaluation of electron density properties. This approach uses a simple exponential function derived from comparison of electron density, gained from high-resolution X-ray crystallography, and distance to atomic nucleus what allows calculating density distribution in time-saving manner and gives results which are very close to experimental ones. As input data AlteQ accepts atomic coordinates of isolated molecules or molecular ensembles (for instance, protein–protein complexes or complexes of small molecules with proteins, etc.). Using AlteQ characteristics we have developed regression models predicting Cioslowski–Mixon bond order (CMBO) indexes (J Am Chem Soc 113(42):4142–4145, 1991). The models are characterized by high correlation coefficients lying in the range from 0.844 to 0.988 dependently on the type of covalent bond, thereby providing a bonding quantification that is in reasonable agreement with that obtained by orbital theory. Comparative analysis of CMBOs approximated using topological properties of AlteQ and experimental electron densities has shown that the models can be used for fast determination of bond orders directly from X-ray crystallography data and confirmed that AlteQ characteristics can replace experimental ones with satisfactory extent of accuracy.     

Concerning electronegativity as a basic elemental property and why the periodic table is usually represented in its medium form

Electronegativity, described by Linus Pauling described as “The power of an atom in a molecule to attract electrons to itself” (Pauling in The nature of the chemical bond, 3rd edn, Cornell University Press, Ithaca, p 88, 1960), is used to predict bond polarity. There are dozens of methods for empirically quantifying electronegativity including: the original thermochemical technique (Pauling in J Am Chem Soc 54:3570–3582, 1932), numerical averaging of the ionisation potential and electron affinity (Mulliken in J Chem Phys 2:782–784, 1934), effective nuclear charge and covalent radius analysis (Sanderson in J Chem Phys 23:2467, 1955) and the averaged successive ionisation energies of an element’s valence electrons (Martynov and Batsanov in Zhurnal Neorganicheskoi Khimii 5:3171–3175, 1980), etc. Indeed, there are such strong correlations between numerous atomic parameters—physical and chemical—that the term “electronegativity” integrates them into a single dimensionless number between 0.78 and 4.00 that can be used to predict/describe/model much of an element’s physical character and chemical behaviour. The design of the common and popular medium form of the periodic table is in large part determined by four quantum numbers and four associated rules. However, adding electronegativity completes the construction so that it displays the multi-parameter periodic law operating in two dimensions, down the groups and across the periods, with minimal ambiguity.     

The principal measure and distributional (p, q)-chaos of a coupled lattice system related with Belusov–Zhabotinskii reaction

García Guirao and Lampart (J Math Chem 48:66–71, 2010; J Math Chem 2 48:159–164, 2010) said that for non-zero couplings constant, the lattice dynamical system is more complicated. Motivated by this, in this paper, we prove that this coupled lattice system is distributionally (p, q)-chaotic for any pair 0?≤ p?≤ q?≤ 1 and its principal measure is not less than ${\frac{2}{3} + \sum_{n=2}^{\infty} \frac{1}{n} \frac{2^{n-1}}{(2^{n}+1)(2^{n-1}+1)}}$ for coupling constant ${0 < \epsilon < 1}$ .     

On-line optimizing control of the intermittent simulated moving bed process

The I-SMB process is one of the modifications to the standard SMB process that has been demonstrated both theoretically and experimentally to exhibit rather competitive performance (Katsuo and Mazzotti in J Chromatogr A 1217:1354, 2010a, 3067, 2010b; Katsuo et al. in J Chromatogr A 1218:9345, 2011). This work aims at showing that also the I-SMB process can be controlled and optimized by using the optimizing on-line controller developed at ETH Zurich for the standard SMB process (Erdem et al. in Ind Eng Chem Res 43:405, 2004a, 3895, 2004b; Grossmann et al. in Adsorption 14:423, 2008, AIChE J 54:1942008). This is achieved by using a virtual I-SMB unit based on a detailed model of the process; past experience with the on-line controller shows that the controller’s behavior on a virtual platform is essentially the same as in laboratory experiments.     

In-situ imaging sensors for bioprocess monitoring: state of the art

Over the last two decades, more and more applications of sophisticated sensor technology have been described in the literature on upstreaming and downstreaming for biotechnological processes (Middendorf et al. J Biotechnol 31:395–403, 1993; Lausch et al. J Chromatogr A 654:190–195, 1993; Scheper et al. Ann NY Acad Sci 506:431–445, 1987), in order to improve the quality and stability of these processes. Generally, biotechnological processes consist of complex three-phase systems—the cells (solid phase) are suspended in medium (liquid phase) and will be streamed by a gas phase. The chemical analysis of such processes has to observe all three phases. Furthermore, the bioanalytical processes used must monitor physical process values (e.g. temperature, shear force), chemical process values (e.g. pH), and biological process values (metabolic state of cell, morphology). In particular, for monitoring and estimation of relevant biological process variables, image-based inline sensors are used increasingly. Of special interest are sensors which can be installed in a bioreactor as sensor probes (e.g. pH probe). The cultivation medium is directly monitored in the process without any need for withdrawal of samples or bypassing. Important variables for the control of such processes are cell count, cell-size distribution (CSD), and the morphology of cells (Höpfner et al. Bioprocess Biosyst Eng 33:247–256, 2010). A major impetus for the development of these image-based techniques is the process analytical technology (PAT) initiative of the US Food and Drug Administration (FDA) (Scheper et al. Anal Chim Acta 163:111–118, 1984; Reardon and Scheper 1995; Schügerl et al. Trends Biotechnol 4:11–15, 1986). This contribution gives an overview of non-invasive, image-based, in-situ systems and their applications. The main focus is directed at the wide application area of in-situ microscopes. These inline image analysis systems enable the determination of indirect and direct cell variables in real time without sampling, but also have application potential in crystallization, material analysis, polymer research, and the petrochemical industry.

The Homfly polynomial of double crossover links

Motivated by double crossover DNA polyhedra (He et al. in Nature 452:198, 2008; Lin et al. in Biochemistry 48:1663, 2009; Zhang et al. in J Am Chem Soc 131:1413, 2009; Zhang et al. in Proc Natl Acad Sci USA 105:10665, 2008; He et al. in Angew Chem Int Ed 49:748, )2010, in this paper, we construct a new type of link, called the double crossover link, formed by utilizing the “ $n$ -point star” to cover each vertex of a connected graph $G$ . The double crossover link can be used to characterize the topological properties of double crossover DNA polyhedra. We show that the Homfly polynomial of the double crossover link can be obtained from the chain polynomial of the truncated graph of $G$ with two distinct labels. As an application, by using computer algebra (Maple) techniques, the Homfly polynomial of a double crossover tetrahedral link is obtained. Our result may be used to characterize and analyze the topological structure of DNA polyhedra.     

Drawing information from the ground state G-particle-hole matrix to study electronic excited states

Very recently, we have shown the suitability to combine the G-particle-hole Hypervirial (GHV) equation method (Alcoba et?al. in Int J Quantum Chem 109:3178, 2009) with the Hermitian Operator (HO) method (Bouten et?al. in Nucl Phys A 202:127, 1973) for computing various energy differences of an electronic system spectrum (Valdemoro et?al. in J Math Chem 50:492, 2012). The purpose of this paper is to extend our preliminary studies by applying the combined GHV-HO method to obtain the set of ground and low-lying excited states potential energy curves of several selected electronic systems. The calculations confirm the reliability of the method.     

A variational density matrix approach with nonlocal effective potential

We show that using the Colle–Salvetti correlation-energy functional (Colle and Salvetti in Theoret Chim Acta 37:329, 1975) in the Hartree–Fock-type procedure suggested by Kohn and Sham (Phys Rev 140:A1133PR, 1965), one can calculate quite accurately electronic properties of systems in which the “dynamical” correlation energy is dominant. We compare our results with those obtained by Grabo and Gross (Chem Phys Lett 240:141, 1995) using the optimized effective potential method, and we discuss characteristics and advantages of our procedure.     

Highlights of non-equilibrium,non-isobaric,non-isothermal desorption of nitrogen from a LiX zeolite column by rapid pressure reduction and rapid purge by oxygen

The effects of adsorption kinetics, column pressure drop, gas phase mass and heat dispersions, gas–solid heat transfer resistance, and adsorber adiabaticity on desorption of N₂ from a LiX zeolite column by O₂ purge as well as pressurization–depressurization of the column using pure N₂ were recently studied using a numerical model of these processes [Chai et al. in Ind Eng Chem Res 50:8703, 2011, Chai et al. in Adsorption 18:87, 2012, Chai et al. in AIChE J 59:365 2013; Rama Rao et al. in Adsorption 2013]. The role of adsorbent particle size and column length to diameter ratio in determining the durations and efficiency of these processes were also investigated. These studies revealed several important limiting and optimum conditions for optimum operation of these processes which can be useful in design of a practical rapid pressure swing adsorption (RPSA) process for medical oxygen concentrator (MOC) application. The purpose of this short review article is to consolidate and re-emphasize these important results in a single article to be used as a guideline for design of a RPSA-MOC unit.     

The development of radioactive glass surrogates for fallout debris

The production of glass that emulates fallout is desired by the nuclear forensics community for training and measurement exercises. The composition of nuclear fallout is complex, with widely varying isotopic compositions (Fahey et al., Proc Natl Acad Sci USA 107(47):20207–20212, 2010; Bellucci et al., Anal Chem 85:7588–7593, 2013; Wallace et al., J Radioanal Nucl Chem, 2013; Belloni et al., J Environ Radioact 102:852–862, 2011; Freiling, Science 139:1058–1059, 1963; Science 133:1991–1999, 1961; Bunney and Sam Government Report: Naval Ordinance Laboratory, White Oak, 1971). As the gaseous cloud traverses from hotter to cooler regions of the atmosphere, the processes of condensation and nucleation entrain environmental materials, vaporized nuclear materials and fission products. The elemental and isotopic composition of the fission products is altered due to chemical fractionation (i.e. the fission product composition that would be expected from fission of the original nuclear material is altered by differences in condensation rates of the elements); the fallout may be enriched or depleted in volatile or refractory fission products. This paper describes preliminary work to synthesize, irradiate and fractionate the fission product content of irradiated particulate glass using a thermal distillation 2 h after irradiation. The glass was synthesized using a solution-based polymerization of tetraethyl orthosilicate. (Izrael, Radioactive fallout after nuclear explosions and accidents, 2002) Uranium was incorporated into the glass particulate at trace concentrations during polymerization. The particulate was subjected to a short thermal neutron irradiation then heated to 1,273 K approximately 2 h after the end of irradiation. Fission products of ^{133, 134, 135}I, ^{132, 134}Te, ¹³⁵Xe, ¹³⁸Cs and ^{91, 92}Sr were observed to be distilled from the particulate. The results of these preliminary studies are discussed.