The principal measure and distributional (p, q)-chaos of a coupled lattice system with coupling constant \varepsilon =1 related with Belusov–Zhabotinskii reaction

We consider the following system coming from a lattice dynamical system stated by Kaneko (Phys Rev Lett, 65:1391–1394, 1990) which is related to the Belusov–Zhabotinskii reaction: $$\begin{aligned} x_{n}^{m+1}=(1-\varepsilon )f\left( x_{n}^{m}\right) +\frac{1}{2}\varepsilon \left[ f(x_{n-1}^{m})+f\left( x_{n+1}^{m}\right) \right] , \end{aligned}$$ where $m$ is discrete time index, $n$ is lattice side index with system size $L$ (i.e., $n=1, 2, \ldots , L$ ), $\varepsilon \ge 0$ is coupling constant, and $f(x)$ is the unimodal map on $I$ (i.e., $f(0)=f(1)=0$ , and $f$ has unique critical point $c$ with $0<c<1$ and $f(c)=1$ ). In this paper, we prove that for coupling constant $\varepsilon =1$ , this CML (Coupled Map Lattice) system is distributionally $(p, q)$ -chaotic for any $p, q\in [0, 1]$ with $p\le q$ , and that its principal measure is not less than $\mu _{p}(f)$ . Consequently, the principal measure of this system 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 $\varepsilon =1$ and the tent map $\Lambda $ defined by $\Lambda (x)=1-|1-2x|, x\in [0, 1]$ . So, our results complement the results of Wu and Zhu (J Math Chem, 50:2439–2445, 2012).     

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

This paper is concerned with the following system which comes from a lattice dynamical system stated by Kaneko in (Phys Rev Lett 65:1391–1394, 1990) and is related to the Belusov–Zhabotinskii reaction: $$\begin{aligned} x_{n}^{m+1}=(1-\varepsilon )f(x_{n}^{m})+\frac{1}{2}\varepsilon \left[ f(x_{n-1}^{m})+f(x_{n+1}^{m})\right] , \end{aligned}$$ x n m + 1 = ( 1 ? ε ) f ( x n m ) + 1 2 ε [ f ( x n ? 1 m ) + f ( x n + 1 m ) ] , where $m$ m is discrete time index, $n$ n is lattice side index with system size $L$ L (i.e., $n=1, 2, \ldots , L$ n = 1 , 2 , … , L ), $\varepsilon $ ε is coupling constant, and $f(x)$ f ( x ) is the unimodal map on $I$ I (i.e., $f(0)=f(1)=0$ f ( 0 ) = f ( 1 ) = 0 and $f$ f has unique critical point $c$ c with $0<c<1$ 0 < c < 1 and $f(c)=1$ f ( c ) = 1 ). It is proved that for coupling constant $\varepsilon =1$ ε = 1 , this CML (Coupled Map Lattice) system is chaotic in the sense of Li–Yorke for each unimodal selfmap on the interval $I=[0, 1]$ I = [ 0 , 1 ] .     

Towards a definition of almost–equienergetic graphs

The energy $E(G)$ of a graph $G$ , a quantity closely related to total $\pi $ -electron energy, is equal to the sum of absolute values of the eigenvalues of $G$ . Two graphs $G_a$ and $G_b$ are said to be equienergetic if $E(G_a)=E(G_b)$ . In 2009 it was discovered that there are pairs of graphs for which the difference $E(G_a)-E(G_b)$ is non-zero, but very small. Such pairs of graphs were referred to as almost equienergetic, but a precise criterion for almost–equienergeticity was not given. We now fill this gap.     

A transformed time-dependent Michaelis–Menten enzymatic reaction model and its asymptotic stability

The dynamic form of the Michaelis–Menten enzymatic reaction equations provide a time-dependent model in which a substrate $S$ reacts with an enzyme $E$ to form a complex $C$ which in turn is converted into a product $P$ and the enzyme $E$ . In the present paper, we show that this system of four nonlinear equations can be reduced to a single nonlinear differential equation, which is simpler to solve numerically than the system of four equations. Applying the Lyapunov stability theory, we prove that the non-zero equilibrium for this equation is globally asymptotically stable, and hence that the non-zero steady-state solution for the full Michaelis–Menten enzymatic reaction model is globally asymptotically stable for all values of the model parameters. As such, the steady-state solutions considered in the literature are stable. We finally discuss properties of the numerical solutions to the dynamic Michaelis–Menten enzymatic reaction model, and show that at small and large time scales the solutions may be approximated analytically.     

Study of Malachite Green Fading in Water–Ethanol–Ethylene Glycol Ternary Mixtures

The rate constant of malachite green (MG⁺) alkaline fading was measured in water–ethanol–ethylene glycol ternary mixtures. This reaction was studied under pseudo-first-order conditions at 283–303 K. In each series of experiments, the concentration of ethanol was kept constant and the concentration of ethylene glycol was changed. It was shown that due to hydrogen bonding and hydrophobic interaction between MG⁺ and alcohol molecules the observed reaction rate constant, $ k_{\text{obs}} $ , increased in the water–ethanol–ethylene glycol ternary mixtures. The fundamental rate constants of MG⁺ fading in these solutions ( $ k_{1} $ , $ k_{ - 1} $ and $ k_{2} $ ) were obtained by the SESMORTAC model. Analysis of $ k_{1} $ and $ k_{2} $ values in solutions containing constant ethanol concentrations show that in low concentrations of ethylene glycol, hydrogen bonding formed between ethanol and ethylene glycol molecules and in high concentrations of ethylene glycol, ethanol as a solvent for ethylene glycol affected the reaction rate.     

A note on the principal measure and distributional ({\varvec{p, q}})-chaos of a coupled lattice system related with Belusov–Zhabotinskii reaction

García Guirao and Lampart in (J Math Chem 48:159–164, 2010) presented a lattice dynamical system stated by Kaneko in (Phys Rev Lett 65:1391–1394, 1990) which is related to the Belusov–Zhabotinskii reaction. In this paper, we prove that for any non-zero coupling constant $\varepsilon \in (0, 1)$ , this coupled map lattice system is distributionally $(p, q)$ -chaotic for any pair $0\le p\le q\le 1$ , and that its principal measure is not less than $(1-\varepsilon )\mu _{p}(f)$ . Consequently, the principal measure of this system is not less than $$\begin{aligned} (1-\varepsilon )\left( \frac{2}{3}+\sum \limits _{n=2}^{\infty }\frac{1}{n}\frac{2^{n-1}}{(2^{n}+1) (2^{n-1}+1)}\right) \end{aligned}$$ for any non-zero coupling constant $\varepsilon \in (0, 1)$ and the tent map $\Lambda $ defined by $$\begin{aligned} \Lambda (x)=1-|1-2x|,\quad x\in [0, 1]. \end{aligned}$$      

CORAL: QSPRs of enthalpies of formation of organometallic compounds

Available on the Internet the CORAL software gives reasonable good prediction for standard enthalpy of formation for selected organometallic compounds (n = 132). The approach is tested using five random splits of the considered data into the sub-training set (n = 32–49), calibration set (n = 36–51), test set (n = 10–29), and the validation set (n = 22–41). Compounds of the validation set are not involved in building up the models. The average statistical quality of prediction is the following: correlation coefficient ( $\overline{R^{2}} )$ R 2 ¯ ) $0.991\pm 0.005$ 0.991 ± 0.005 and standard error of estimation ( $\overline{s} )$ s ¯ ) $22.9 \pm 5.6$ 22.9 ± 5.6  kJ/mol.     

Mechanistic description of survival of irradiated cells: repair kinetics in Padé linear-quadratic or differential Michaelis–Menten model

Biophysical models for repair mechanisms for cell surviving fractions $S_\mathrm{F}(D)$ S F ( D ) after exposure to radiation are studied. The principal focus is on a novel theory, the Padé-linear quadratic (PLQ), or equivalently, the differential Michaelis–Menten radiobiological model, which predicts $S_\mathrm{F}(D)$ S F ( D ) as a function of the absorbed dose $D$ D in the form $S_\mathrm{F}(D)=\exp {\{-(\alpha D+\beta D^2)/(1+\gamma D)\}}$ S F ( D ) = exp { - ( α D + β D 2 ) / ( 1 + γ D ) } , with a clear biological and clinical meaning of the three parameters $\alpha , \beta $ α , β and $\gamma $ γ . It is shown that this functional form in the PLQ model emerges directly from the simultaneous fulfillment of the requirements for the correct asymptotic behaviors of the repair function at low and high doses. Moreover, this automatically secures the purely exponential cell kill modes at both small and large $D$ D , as also encountered in the corresponding experimental data for cell surviving fractions. Further, it is demonstrated that the PLQ-based repair function, given by a rectangular hyperbola, coincides with the reaction velocity for enzyme catalysis from the Michaelis–Menten mechanism. This repair velocity is the halved harmonic mean of the low- and high-dose asymptotes of the catalytic repair function. Such circumstances constitute a firm mechanistic basis of the PLQ model, which is shown to exhibit excellent agreement with measurements. Robust applications of the PLQ model are anticipated, especially in hypofractionted radiotherapy, such as stereotactic radiosurgery.     

Equivalence of the deformed Rosen–Morse potential energy model and Tietz potential energy model

By applying the dissociation energy and the equilibrium bond length for a diatomic molecule as explicit parameters, we generate an improved expression for the deformed Rosen–Morse potential energy model. It is found that the deformed Rosen–Morse potential model and the well-known Tietz potential model are the same empirical potential function for diatomic molecules. With the help of the energy spectrum expression of the deformed Rosen–Morse potential model, we obtain exact closed-form expressions of diatomic anharmonicity constants $\omega _e x_e $ ω e x e and $\omega _e y_e $ ω e y e .     

Clar and Fries numbers for benzenoids

A Kekulé structure of a benzenoid or a fullerene $\Gamma $ Γ is a set of edges $K$ K such that each vertex of $\Gamma $ Γ is incident with exactly one edge in $K$ K . The set of faces in $\Gamma $ Γ that have exactly three edges in $K$ K are called the benzene faces of $K$ K . The Fries number of $\Gamma $ Γ is the maximum number of benzene faces over all possible Kekulé structures for $\Gamma $ Γ . The Clar number is the maximum number of independent benzene faces over all possible Kekulé structures for $\Gamma $ Γ . It is often assumed, but never proved, that some set of independent benzene faces giving the Clar number is a subset of a set of benzene faces giving the Fries number. In Hartung (The Clar structure of fullerenes, Ph.D. Dissertation. Syracuse University, 2012) it is shown that this assumption is false for a large class of fullerenes. In this paper, we prove that this assumption is valid for a large a class of benzenoids.     

Ro-vibrational spectroscopy of molecules represented by a Tietz–Hua oscillator potential

Accurate low and high-lying bound states of Tietz–Hua oscillator potential are presented. The radial Schrödinger equation is solved efficiently by means of the generalized pseudospectral method that enables optimal spatial discreti zation. Both $\ell =0$ and rotational states are considered. Ro-vibrational levels of six diatomic molecules viz., H $_2$ , HF, N $_2$ , NO, O $_2$ , O $_2^+$ are obtained with good accuracy. Most of the states are reported here for the first time. A detailed analysis of variation of eigenvalues with $n, \ell $ quantum numbers is made. Results are compared with literature data, wherever possible. These are also briefly contrasted with the Morse potential results.     

Compound-specific isotope analysis of benzotriazole and its derivatives

Compound-specific isotope analysis (CSIA) is an important tool for the identification of contaminant sources and transformation pathways, but it is rarely applied to emerging aquatic micropollutants owing to a series of instrumental challenges. Using four different benzotriazole corrosion inhibitors and its derivatives as examples, we obtained evidence that formation of organometallic complexes of benzotriazoles with parts of the instrumentation impedes isotope analysis. Therefore, we propose two strategies for accurate $\delta^{13}$ C and $\delta^{15}$ N measurements of polar organic micropollutants by gas chromatography isotope ratio mass spectrometry (GC/IRMS). Our first approach avoids metallic components and uses a Ni/Pt reactor for benzotriazole combustion while the second is based on the coupling of online methylation to the established GC/IRMS setup. Method detection limits for on-column injection of benzotriazole, as well as its 1-CH $_{3}$ -, 4-CH $_{3}$ -, and 5-CH $_{3}$ -substituted species were 0.1–0.3 mM and 0.1–1.0 mM for δ¹³C and δ¹⁵N analysis respectively, corresponding to injected masses of 0.7–1.8 nmol C and 0.4–3.0 nmol N, respectively. The Ni/Pt reactor showed good precision and was very long-lived ( $>$ 1000 successful measurements). Coupling isotopic analysis to offline solid-phase extraction enabled benzotriazole-CSIA in tap water, wastewater treatment effluent, activated sludge, and in commercial dishwashing products. A comparison of $\delta ^{13}$ C and $\delta ^{15}$ N values from different benzotriazoles and benzotriazole derivatives, both from commercial standards and in dishwashing detergents, reveals the potential application of the proposed method for source apportionment.     

Two-dimensional high-performance liquid chromatographic determination of day–night variation of d-alanine in mammals and factors controlling the circadian changes

d-Alanine (d-Ala) is one of the naturally occurring d-amino acids in mammals, and its amount is known to have characteristic circadian changes. It is a candidate for a novel physiologically active substance and/or a biomarker, and the regulation mechanisms of the intrinsic amounts of d-Ala are expected to be clarified. In the present study, the effects of the possible factors controlling the d-Ala amounts, e.g., diet, d-amino acid oxidase (DAO) and intestinal bacteria, on the day–night changes in the intrinsic d-Ala amounts have been investigated using a highly sensitive and selective two-dimensional high-performance liquid chromatographic system combining a reversed-phase column and an enantioselective column. The circadian rhythm was not changed under fasting conditions. In the mice lacking d-amino acid oxidase activity (ddY/DAO^- mice), clear day–night changes were still observed, suggesting that the factors controlling the d-Ala rhythm were not their food and DAO activity. On the other hand, in the germ-free mice, quite low amounts of d-Ala were detected compared with those in the control mice, indicating that the main origin of d-Ala in the mice is intestinal bacteria. Because the d-Ala amounts in the digesta containing intestinal bacteria did not show the day–night changes, the controlling factor of the circadian changes of the d-Ala amount was suggested to be the intestinal absorption.     

A boundary-driven reaction front

A model reaction scheme in which two species $A$ and $B$ react to form an inert product is considered, with the possible linear decay of $A$ to a further inert prduct also included. The reaction between $A$ and $B$ is maintained by the input of $A$ from the boundary which keeps $A$ at a constant concentration. The cases when $B$ is immobile or free to diffuse are treated. In the former case reaction fronts in $B$ are seen to develop. Large time asymptotic solutions are derived which show that these fronts propagate across the reactor at rates proportional to $t^{1/2}$ or $\log t$ ( $t$ is a dimensionless time) depending on whether the extra decay step is included. A similar situation is seen when $B$ can diffuse when the linear decay step is not present. However, when this extra step is included in the reaction scheme the reaction zone reaches only a finite distance fronm the boundary at large times.     

A new approach to maturity of molecules by rationality of finite groups

These two concepts, maturity in chemistry and rationality in group theory were discovered by a chemist, Fujita. In the present study, we introduce a new approach to maturity and immaturity of simple groups, using the deep theorem (Feit and Seitz in Ill J Math 33:101–131, 1988). Additionally, we prove that 1,3,5-trimethyl-2,4,6-trinitrobenzene are always unmatured and tetra platinum(II) with point group $D_{2n}$ , dihedral group of order $2n$ , is unmatured if $n\ne 1,2,3,4,6$ . Also, we compute integer-valued characters of the simple sporadic group $Ly$ .     

Molecular Interactions in 1-Ethyl-3-Methylimidazolim Tetrafluoroborate + Amide Mixtures: Excess Molar Volumes, Excess Isentropic Compressibilities and Excess Molar Enthalpies

The densities, ρ ₁₂, and speeds of sound, u ₁₂, of 1-ethyl-3-methylimidazolium tetrafluoroborate (1) + N-methylformamide or N,N-dimethylformamide (2) binary mixtures at (293.15. 298.15. 303.15, 308.15 K), and excess molar enthalpies, $ H_{12}^{\text{E}} $ H 12 E , of the same mixtures at 298.15 K have been measured over the entire mole fraction range using a density and sound analyzer (Anton Paar DSA-5000) and a 2-drop microcalorimeter, respectively. Excess molar volume, $ V_{12}^{\text{E}} $ V 12 E , and excess isentropic compressibility, $ \left( {\kappa_{S}^{\text{E}} } \right)_{12} $ ( κ S E ) 12 , values have been calculated by utilizing the measured density and speed of sound data. The observed data have been analyzed in terms of: (i) Graph theory and (ii) the Prigogine–Flory–Patterson theory. Analysis of the $ V_{12}^{\text{E}} $ V 12 E data in terms of Graph theory suggest that: (i) in pure 1-ethyl-3-methylimidazolium tetrafluoroborate, the tetrafluoroborate anion is positioned over the imidazoliun ring and there are interactions between the hydrogen atom of (C–H{edge}) and proton of the –CH₃ group (imidazolium ring) with fluorine atoms of tetrafluoroborate anion, and (ii) (1 + 2) mixtures are characterized by ion–dipole interactions to form a 1:1 molecular complex. Further, the $ V_{12}^{\text{E}} $ V 12 E , $ H_{12}^{\text{E}} $ H 12 E and $ \left( {\kappa_{S}^{\text{E}} } \right)_{12} $ ( κ S E ) 12 values determined from Graph theory compare well with their measured experimental data.     

Non-Hermitian Hydrogen atom

We have constructed a set of non-Hermitian operators that satisfy the commutation relations of the $SO(3)$ SO ( 3 ) -Lie algebra. Using these set of operators we have constructed a non-Hermitian Hamiltonian corresponding to the Hydrogen atom that includes a complex term but with the same spectra as in the Hermitian case. It is also found a non-Hermitian Runge–Lenz vector that represents a conserved quantity. In this way, we obtain a set of non-Hermitian operators that satisfy the commutation relations of the $SO(4)$ SO ( 4 ) -Lie algebra.     

An IEF-PCM study of solvent effects on the Faraday {\mathcal{B}} term of MCD

We present the first theoretical investigation of solvent effects on the Faraday ${\mathcal{B}}$ term of magnetic circular dichroism (MCD) at the density–functional level of theory. In our model, the solvent is described by the polarizable continuum model in its integral-equation formulation. We present the extensions required for including electron correlation effects using density–functional theory (DFT) as well as the necessary extensions for including the effects of a dielectric continuum. The new code is applied to the study of the Faraday ${\mathcal{B}}$ term of MCD in a series of benzoquinones. It is demonstrated that electron correlation effects, as described by DFT, are essential in order to recover the experimentally observed signs of the ${\mathcal{B}}$ term. Dielectric continuum effects increase, in general, the magnitude of the ${\mathcal{B}}$ term, leading to an overestimation of the experimental observations in most cases.     

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 ${0 < \epsilon <1 }$ .