Journal of Mathematical Chemistry最新文献

The main objective of this research is to develop and analyze high-order symmetric Gauss-type exponential collocation time-stepping methods for solving systems of nonlinear first-order partial differential equations (PDEs). Initially, the nonlinear PDEs are reformulated as an abstract Hamiltonian ordinary differential equation (ODE) system in an appropriate infinite-dimensional function space. Subsequently, the Gauss-type exponential collocation time integrators are derived. The symmetry, local error bounds and nonlinear stability of the proposed time integrators are rigorously analysed in details. Furthermore, the rigourous convergence analysis demonstrates that Gauss-type exponential collocation time integrators can achieve superconvergence. Numerical experiments verify our theoretical analysis results, and demonstrate the remarkable superiority in comparison with the traditional temporal integration methods.

This study presents a novel low-order Adams–Bashforth–Moulton predictor–corrector algorithm. This new approach was able to incorporate any linear combination of the functions (left[ 1, , x, , x^2, , e^{I, v , x} right] ). Stability zones are established for the novel approach. Using cases from chemistry and other fields, we test how well the recently suggested technique works.

We revisit the quantum-mechanical two-dimensional hydrogen atom with an electric field confined to a circular box of impenetrable wall. In order to obtain the energy spectrum we resort to the Rayleigh–Ritz method with a polynomial basis sets. We discuss the limits of large and small box radius and the symmetry of the solutions of the Schrödinger equation. An interesting feature of the model is the appearance of accidental degeneracy and the splitting of degenerate energy levels due to the presence of the electric field.

Cage graphs are vertex-regular graphs with a given girth, and they find several chemical and biological applications including the representations of isomerization reaction pathways and other chemical and biological networks. A (d, g)- cage contains all vertices with the same degree d and has a girth g. In chemical applications trivalent-cages play especially important roles, as exemplified by the applications to dynamic stereochemistry. Several of the cage graphs exhibit very high order of symmetries, and hence highly degenerate spectra and often integral spectra. The cage graphs have a long and rich history of research-span tracing back to Kárteszi, Sachs and Erdös. We obtain fully expanded spectral polynomials, graph spectra and a number of distance and degree based descriptors, graph energies and entropies of several mathematically and chemically interesting cages. The spectral polynomials of the cages are computed through powerful bit-manipulation algorithms. We have considered both vertex-transitive cages and cages with multiple or single automorphic vertex equivalence classes of vertices. Computations were carried out using high degree of precision to enumerate the coefficients of the spectral polynomials and other properties. The mathematical properties and the coefficients in the polynomials were further dissected and analysed to provide structural interpretations.

The realm of the current study is ovarian magnetic resonance spectroscopy (MRS). Presented are the selected recent advances in the Padé-based signal processing by shape estimations alone. The goal is to substantially improve extraction of quantitative information by sole reliance upon non-parametric estimations of total shape spectra (envelopes) from encoded time signals. The task is to resolve the given envelope into its true partial spectra (components) without solving the quantification problem (i.e. no polynomial rooting, etc.). The rescue is in derivative quantitative shape estimations void of fitting. Splitting apart an envelope into the genuine components amounts to quantification. With any quadrature rule, integrations of the reconstructed well-isolated unstructured derivative lineshapes and their power spectra determine the peak areas and peak widths, respectively. Metabolite concentrations ensue thereby as a key diagnostic information for recognized and potential cancer biomarkers alike. Special attention is drawn to abundant non-derivative singlet-appearing resonances that can contain sub-peaks in derivative lineshapes. Failure to detect such occurrences compromises the critical decision-making (normal vs. diseased tissues or biofluids) in the clinic. The salient illustrations are reported for benign and malignant tumors from human ovarian cyst fluid samples.

The topic of aromaticity of polycyclic assembly of hexagons has been of considerable interest over the decades. Polyhex carbon nanotubes in different topologies are especially intriguing from the standpoint of aromaticity, Kekulé structures, Dewar structure counts and various polynomials pertinent to these structures. In this study we juxtapose the various novel aromatic measures of zigzag versus armchair polyhex single-walled carbon nanotubes as a function of their tube lengths and circumference. We have computed the matching, spectral and delta polynomials of the two topologies of these tubes as a function of their lengths and circumferences. Our computations reveal that for a single-walled tube, the armchair is more aromatic than zigzag by all measures, and as the tube length increases aromaticity increases for both configurations. In contrast, as the tube circumference increases the aromaticity increases for the zigzag while a less pronounced opposite trend is exhibited by the armchair. There is a dramatic odd-even alternation in zigzag tubes with even parameter exhibiting zero gap, and for both tubes odd parameters exhibit perfect square constant coefficients in the spectral polynomials correlating with the square of the Kekulé structure counts. The armchair tube exhibits a much greater number of Dewar and Kekulé structures for any given set of parameters of the tube compared to the zigzag tubes confirming a greater aromaticity of the armchair tube. For example, the armchair [5,7] tube exhibits K = 65,445 and DS = 12,001,780 compared to the zigzag[5,7] with K = 128 and DS = 1,071,345. On the other hand, the zigzag nanotubes exhibit greater entropies compared to the armchair nanotubes. Applications of the combinatorial and computational techniques to various parameters related to the aromaticity and stability of different topologies of carbon nanotubes are considered.

This work belongs to the field of chemical kinetics and is devoted to the frequency factor of particle collisions occurring in an inert gaseous or liquid medium. The objective of the research is to account for the influence of the inert medium on the rate of a chemical reaction. This was achieved using methods of traditional chemical kinetics and advanced mathematics (probability theory and mathematical analysis). By employing a special mathematical object – the Wiener sausage – an estimate of the probability of collision between two particles over time was obtained. In the case of multiple particles, the frequency factor was interpreted as a quantity proportional the collision probability. The new form of the Arrhenius equation derived in this study includes three terms on the pre-exponential factor, which adds novelty and significance to the research. Additionally, the paper provides a physical interpretation of each term in the newly obtained frequency factor formula and successfully validates the correctness of the new Arrhenius equation using a specific example for comparison with experimental results.

In this paper, a novel high-order, mass and energy-conserving scheme is proposed for the regularized logarithmic Schrödinger equation. Based on the idea of the supplementary variable method (SVM), we firstly reformulate the original system into an equivalent form by introducing two supplementary variables, and the resulting SVM reformulation is then discretized by applying a high-order prediction-correction method in time and a Fourier pseudo-spectral method in space, respectively. The newly developed scheme can produce numerical solutions along which the mass and original energy are precisely conserved, as is the case with the analytical solution. Additionally, it is extremely efficient in the sense that only requires solving a constant-coefficient linear systems plus two algebraic equations, which can be efficiently solved by the Newton iteration at every time step. Numerical experiments are presented to confirm the accuracy and structure-preserving properties of the new scheme.

We propose a new mathematical model of the Briggs–Rauscher reaction. This is an oscillatory phenomenon which is characterised by fluctuations in the concentrations of the various chemicals involved. A well-regarded existing model involves a complex reaction mechanism described by 15 differential equations. We derive a novel approximate mathematical model that consists only of three equations, for the concentrations of iodous acid, iodide, and molecular iodine. We demonstrate that this three-variable approximation is nevertheless in good agreement with the predictions of far more elaborate models, and it offers the possibility of yielding to detailed mathematical analysis not available with more complex models. We show that our novel three-variable description is in excellent accord with previously-reported experimental work. It is able to reproduce key details of the observed periodic oscillations, including their period and amplitude and precise features of their behaviour with time.