A New Eighth-order A-stable Method for Solving Differential Systems Arising in Chemical Reactions

Implicit Runge–Kutta methods are successful algorithms for the numerical solution of stiff differential equations, as they usually appear in chemical reactions. This article describes the construction of a particular implicit method based on internal stages obtained from certain Chebyshev collocation points. The resulting method has algebraic order 8 and A-stability characteristic. An embedding technique using the Runge–Kutta method and a linear multistep one is provided in order to change the step size. Numerical experiments illustrate the behaviour of the new method, showing that it may reach great accuracy and be competitive with other well-known codes.     

Crystal and molecular structure of (r-2, c-4)-3-benzyl-2,4,5,5-tetraphenyl1,3-thiazolidine, intramolecular C–HS hydrogen bonds

The crystal and molecular structures of (r-2, c-4)-3-benzyl-2,4,5,5-tetraphenyl-1,3-thiazolidine are investigated showing the existence of C(sp²)–HS and C(sp²)–HN intramolecular contacts. The analysis of geometrical parameters shows that C–HS contacts may be treated as hydrogen bonds but C–HN do not fulfil the geometrical criteria of the existence of H-bonds. The B3LYP/6-311+G* single point calculations were performed to obtain wave functions applied later for ‘atoms in molecules’ (AIM) study. The analysis of bond critical points based on the Bader theory (AIM) supports the existence of intramolecular C–HS H-bonds.     

On the number of invariant measures for higher-dimensional chaotic transformations

LetS be a bounded region inR ^N and letP={S _l } _i=1 ^m be a partition ofS into a finite number of closed subsets having piecewiseC ² boundaries of finite (N–1)-dimensional measure. Let :SS be piecewiseC ² onP and expanding in the sense that there exists 0<<1 such that for anyi=1,2,...,m, DT _i ^–1<, whereDT _i ^–1 is the derivative matrix ofT _i ^–1 and · is the Euclidean matrix norm. We prove that for some classes of such mappings, for example, Jabtonski transformations or convexity-preserving transformations, the number of crossing points constitutes a bound for the number of ergodic absolutely continuous -invariant measures. We give examples showing that in general the simple bound of one-dimensional dynamics cannot be generalized to higher dimensions. In fact, we show that it is possible to construct piecewise expandingC ² transformations on a fixed partition with a finite number of elements but which have an arbitrarily large number of ergodic, absolutely continuous invariant measures.     

Stationary points on the aminomethanol potential energy surface

Summary In this work we study surface fitting equations for a rigid rotor model of aminomethanol. The energies were obtained from the GAUSSIAN88 package using 3-21G bases and fitted on a least square equation, thus generating a Fourier series expansion of the energy as a function of two dihedral angles. The dihedral angles chosen are those that represent rotation around the C-O and N-C axes in the first case, and rotation around C-O and inversion around the amino group in the second case. Results indicate that the hydroxyl hydrogen is subject to almost free rotation around the C-O axis. Further fully relaxed 6-31G* calculations were performed in order to qualify the results obtained for the rigid rotor model.     

Isoelectric point separation of proteins by capillary pH-gradient ion-exchange chromatography

In the present work, isoelectric point (pl) separation of proteins by pH-gradient ion-exchange chromatography (IEC) on packed capillary columns is demonstrated. The development of a miniaturized flow-through pH probe for reliable pH monitoring of the column effluent, which was an important technical challenge for adapting this technique to capillary dimensions, was solved by designing a low microliter per minute flow rate housing to a commercially available micro pH probe. Highly linear outlet pH-gradients within the pH range 8.5-4.0 were obtained when applying simple inexpensive buffers consisting solely of piperazine, N-methylpiperazine and imidazole on 10 cm x 0.32 mm i.d. fused silica capillaries packed with anion-exchange poly(styrene divinylbenzene)-based macroporous materials, i.e. 10 microm Mono P from Amersham Biosciences and 10 microm PL-SAX from PolymerLabs. Furthermore, when using a pH-gradient from 6.8 to 4.3, both columns were able to baseline separate the A and B genetic variants of beta-lactoglobulin, which differ with two amino acid residues only, but the PL-SAX column provided almost a two-fold decrease in peak widths compared to the Mono P column. The influence of varying the buffer concentration, injection volume and column temperature on the peak widths and resolution of the beta-lactoglobulins was investigated, e.g. a 100 microl sample of dilute beta-lactoglobulins was injected directly on the column with practically no increase in peak width as compared to what obtained with conventional injection volumes. Finally, a pH-gradient from 6.8 to 4.3 was used to separate proteins in skimmed bovine milk on the PL-SAX column. The milk was simply diluted 1:10 (v/v) with water and filtrated before injection.     

Interactions in aqueous nonelectrolyte systems. Gibbs energy of interaction of the ether group with the hydroxyl group and the amide group

Freezing temperatures of dilute aqueous solutions of equimolar mixtures of 1,3,5-trioxane with myo-inositol, d-mannitol, cyclohexanol, formamide, and acetamide, and 1,4-dioxane with myo-inositol, d-mannitol, formamide, and acetamide have been measured. These data yield pairwise Gibbs energies of interactions between the molecules in an aqueous solution. Using the group additivity principle, the results also yield the pairwise functional group Gibbs energies of interaction for the ether group with the hydroxyl and amide group. These results have been combined with all available data from the literature to yield the Gibbs energy and enthalpy of interaction of amides, ethers, alcohols, and saccharides in aqueous solution.To whom correspondence should be addressed.     

Freezing temperatures and enthalpies of dilution of aqueous solutions of amides. Gibbs energies and enthalpies of interaction of the N,N-dimethylamide group in aqueous solutions

Enthalpies of dilution, freezing temperatures, and densities of aqueous solutions of N,N-dimethylacetamide and N,N-dimethylpropionamide have been measured. Freezing temperatures of dilute aqueous solutions of formamide and N,N-dimethylformamide have also been measured. These data yield the pairwise molecular Gibbs energies and enthalpies of interaction: these have been treated according to a group additivity principle to give pairwise functional group Gibbs energies and enthalpies of interaction. The results indicate that substitution on the amide nitrogen may increase the Gibbs energy and enthalpy of interaction of the amide group with itself in an aqueous environment but the effect if present is small.     

Effects of inorganic salts on the properties of aqueous poly(vinylpyrrolidone) solutions

Cloud point curves and temperatures have been determined for aqueous solutions of poly(vinylpyrrolidone) at several concentrations for a variety of inorganic salts (phosphates, monohydrogen phosphates, sulfates, carbonates, dihydrogen phosphates and fluorides). The resulting dependency of the critical temperatures (mostly between 289 and 350 K) on the molar concentration can be expressed as sequences showing the decreasing effect of anion species or cation species in salting out the polymer. The decreasing order of effectiveness of the anions in reducing the temperatures is PO ₄ ^3– >HPO ₄ ^2– >SO ₄ ^2– CO ₃ ^2– >H₂PO ₄ ^– >F^–. The order for cation is Na⁺>K⁺. The changes brought about in temperatures by the salts were found to be the results of the changes taking place in the hydrophilic and hydrophobic interactions among polymer, solvent and additive salts and of the change of water structure by structure making or structure breaking ions, and of the influence of salts on the hydration sheath of the polymer.Deceased     

Bifurcation of reaction pathways: the set of valley ridge inflection points of a simple three-dimensional potential energy surface

This paper serves for the better understanding of the branching phenomenon of reaction paths of potential energy hypersurfaces in more than two dimensions. We apply the recently proposed reduced gradient following (RGF) method for the analysis of potential energy hypersurfaces having valley-ridge inflection (VRI) points. VRI points indicate the region of possible reaction path bifurcation. The relation between RGF and the so-called global Newton search for stationary points (Branin method) is shown. Using a 3D polynomial test surface, a whole 1D manifold of VRI points is obtained. Its relation to RGF curves, steepest descent and gradient extremals is discussed as well as the relation of the VRI manifold to bifurcation points of these curves. Received: 8 July 1998 / Accepted: 24 August 1998 / Published online: 23 November 1998     

Control problems with random and progressively known targets

The situation considered is of optimally controlling a deterministic system from a given state to an initially unknown targety in a fixed time interval [T ₀,T]. The target will be certainly known at a random time in [T ₀,T]. The controller knows the distributions ofy and . We derive the Bellman equation for the problem, prove a verification theorem for it, and demonstrate how the distribution influences the optimal control. We show that, in the linear-quadratic case, the optimal control is given by a feedback law that does not depend on the distribution of .The author wishes to express his gratitude to Prof. Wendell H. Fleming and Prof. Steven Orey for fruitful discussion concerning this work.This research was supported in part by the Institute for Mathematics and Its Applications with funds provided by the National Science Foundation.