Journal of Mathematical Chemistry最新文献

We apply the well known Rayleigh–Ritz method (RRM) to the projection of a Hamiltonian operator chosen recently for the extension of the Rayleigh–Ritz variational principle to ensemble states. By means of a toy model we show that the RRM eigenvalues approach to those of the projected Hamiltonian from below in most cases. We also discuss the effect of an energy shift and the projection of the identity operator.

Analytical solutions for the ordinary differential equations are reported for the kinetics of two-step processes for which the later step is a first order reversible process. The earlier step is always irreversible: zeroth order, first order, second order and third order reactions are considered. For the first and second order cases, a qualitative analysis of the kinetic curves was also carried out and the parametric conditions of finding extrema on all the kinetic curves are explored. It is found that the scheme consisting of a second order or mixed second order earlier reaction and a reversible first order later one may feature a reactant with two extrema on its concentration–time trace. In such cases, the first extremum is always a maximum, and the second one is a minimum.

We present a complete characterization of nontrivial local conservation laws for the extended generalized Cahn–Hilliard–Kuramoto–Sivashinsky equation in any space dimension. This equation naturally generalizes the well-known and widely used Cahn–Hilliard and Kuramoto–Sivashinsky equations, which have manifold applications in chemistry, physics, and biology. In particular, we demonstrate that any nontrivial local conservation law of any order for the equation under study is equivalent to a conservation law whose density is linear in the dependent variable with the coefficient at the dependent variable depending at most on the independent variables.

A higher-order time-fractional evolution problems (EPs) with the Caputo time fractional derivative is considered. A weak singularity typically appears close to the initial time ((t=0)) in this problem’s solution, which reduces the accuracy of conventional numerical methods with uniform mesh. The technique of nonuniform mesh based on the solution’s acceptable regularity is a very efficient way to regain precision. In chemistry, these equations are often used to simulate intricate diffusion processes with memory effects, particularly whenever pattern formation, domain wall propagation in liquid crystals are involved. In the current study, we solve a time-fractional fourth-order partial differential equation with non-smooth solutions using the quintic trigonometric B-spline (QTBS) technique with temporally graded mesh. The stability and convergence of the proposed numerical scheme are discussed broadly, which illustrates clearly how the regularity of the solution and the mesh grading affect the order of convergence of the proposed scheme, allowing one to select the most effective mesh grading. The plots and tabulated results of some test problems are displayed to validate the accuracy and efficiency of the scheme using graded mesh.

An exhaustive study is presented in this work to solve a chemical clock reaction model, which has a vital role in chemistry. The non-integer order chemical clock reaction framework in terms of the Caputo operator is discussed in this paper. In this research work, fractional-order chemical clock reaction equations are addressed with the assistance of the Haar wavelet approach. To check that the obtained solutions are correct, the Adams–Bashforth–Moulton method is used. Also, we conducted a comparative study of the outcomes of the chemical clock reaction model with the spectral collocation technique. Further, the Haar wavelet operational matrix is derived to convert the set of differential equation transforms into a set of algebraic equations. This set of complex nonlinear equations is resolved by utilizing MATLAB (2023a). Moreover, the focus lies on the convergent analysis, stability analysis, and the existence and uniqueness of the obtained outcomes. Furthermore, error analysis by contrasting the Haar wavelet technique and the spectral collocation technique is also discussed. This work not only shows the efficiency of the Haar wavelet technique in exactly calculating the dynamics of the chemical clock reaction model but also provides some examination of the chemical clock reaction system. Convergence analysis tells us that (leftVert e_mathfrak {M}(t) rightVert _2 = oleft( frac{1}{mathfrak {M}}right) .) This implies that as ( mathfrak {M} ) increases, the error decreases. Specifically, for ( mathfrak {M} = 8 ), the absolute error is approximately ( 0.125 ), while for ( mathfrak {M} = 16 ) and ( mathfrak {M} = 32 ), the errors reduce to ( 0.0625 ) and ( 0.03125 ), respectively. The error analysis shows that the error between Haar wavelet method and Adams–Bashforth–Moulton method maintain a low error rate, often in the range of ( mathbf {10^{-4}} ) to ( mathbf {10^{-1}} ), whereas the error between Spectral Collocation method and the Adams–Bashforth–Moulton method exhibit higher absolute errors, highlighting accuracy of the Haar wavelet approach. Additionally, the stability of the proposed method is theoretically established, ensuring that the solutions remain bounded within a well-defined range.

No. However, here we show how to generate a metric consistent with the Tanimoto similarity. We also explore new properties of this index, and how it relates to other popular alternatives.

We consider the combinatorial enumeration of random walks on graphs with emphasis on symmetric, vertex-transitive and bipartite generalized Petersen graphs containing up to 720 vertices. We enumerate self-returning and non-returning walks originating from each vertex of graphs using the matrix power algorithms. We formulate the vertex entropies, scaled unit self-return and non-return walk entropies of structures which provide measures for the combinatorial complexity of graphs. We have chosen mathematically and chemically interesting generalized Petersen graphs G(n,k) with floral symmetries, as they find several applications in dynamic stereochemistry and several other fields. These studies reveal several interesting walk patterns and walk sequences for these graphs, and paves the way for statistical studies on these chemically and mathematically interesting graphs. Moreover, walk-based vertex partitions are machine-generated from the enumerated walk n-tuple vectors, although they do not always correlate with the automorphic partitions. Hence the present study attempts to integrate statistical mechanics, graph theory, combinatorial complexity, and symmetry for large molecular and biological networks.

Graphical abstract

In this paper, we consider the determinant evaluation of a generalized comrade matrix based on a novel incomplete block-diagonalization approach which transforms the determinant of the original generalized comrade matrix into the determinants of tridiagonal matrices and comrade matrix with lower-order. Then, a breakdown-free recursive algorithm for computing the determinant of the generalized comrade matrix is proposed. Even though the algorithm is not a symbolic algorithm, it never suffers from breakdown. Furthermore, we propose an explicit formula for the determinant of the generalized comrade matrix with quasi-Toeplitz structure. Some numerical results with simulations in MATLAB implementation are provided to demonstrate the accuracy and effectiveness of the proposed algorithm, and its competitiveness with MATLAB built-in function.

Cyclic tridiagonal matrices, a specific subclass of quasi-tridiagonal matrices, frequently arise in theoretical and computational chemistry. This paper addresses the solution of cyclic tridiagonal linear systems with coefficient matrices that are subdiagonally dominant, superdiagonally dominant and weakly diagonally dominant. For the subdiagonally dominant case, we perform an elementary transformation to convert the matrix into a block 2-by-2 form, then solve the system using block LU factorization. For the superdiagonally dominant and weakly diagonally dominant cases, we extend this approach using block LU factorization and matrix similarity transformations. Our proposed algorithms outperform existing methods in terms of floating-point operations, memory storage, and data transmission. Numerical experiments, implemented in MATLAB, demonstrate the accuracy and efficiency of the proposed algorithms.

In the present note we present some comments on the papers: (1) The use of a multistep, cost-efficient fourteenth-order phase-fitting method to chemistry problems by Rong Xu, Bin Sun, Chia-Liang Lin, T. E. Simos, Journal of Mathematical Chemistry (2024) 62:1781–1807 https://doi.org/10.1007/s10910-024-01623-7 and (2) An effective multistep fourteenth-order phase-fitting approach to solving chemistry problems by Hui Huang, Cheng Liu, Chia-Liang Lin, T. E. Simos, Journal of Mathematical Chemistry (2024) 62:1860–1889, https://doi.org/10.1007/s10910-024-01628-2