Computational and Applied Mathematics最新文献

In this paper, based on Lagrangian functions of the optimization problem we develop a sketch-and-project method for solving the linear matrix equation (AXB = C) by introducing three parameters. A thorough convergence analysis on the proposed method is explored in details. A lower bound on the convergence rate and some convergence conditions are derived. By varying three parameters in the new method and convergence theorems, an array of well-known algorithms and their convergence results are recovered. Finally, numerical experiments are given to illustrate the effectiveness of recovered methods.

Consider a finite field ({mathbb {F}}_{q}) with characteristic p, where G is a crystallographic point group satisfying (p not mid |G|) and (q=p^n). In this paper, we propose studying group codes in the crystallographic point group algebras ({mathbb {F}}_{q}G) for the point groups (C_{2h}), (C_{6v}), and (D_{6h}). We compute the unique (linear and nonlinear) idempotents of ({mathbb {F}}_{q}G) that correspond to the characters of the crystallographic point groups. These idempotents play a crucial role in characterizing the properties of the group codes. Based on the above results, we characterize the minimum distances and dimensions of the group codes. This provides valuable information about the error-correcting capabilities and the amount of information that can be transmitted through these codes. Furthermore, we construct MDS (Maximum Distance Separable) group codes and almost MDS group codes.

This paper introduces a class of two-step collocation methods for the numerical solution of ordinary differential equations. These methods which are equipped with the future point technique and described in two types, to approximate the solution in each step, use the numerical solution in some points in the two previous subintervals as well as in the future subinterval. The superior features of the proposed methods in convergence order and stability in comparison with the similar methods are analyzed. The achieved improvements are verified by giving some numerical experiments.

A shallow-water wave propagation model can be described as a generalized Rosenau–Burgers equation with strong nonlinearity and high-order dispersion terms. In this paper, we propose two generalized high-order (up to eighth-order) compact finite difference schemes for solving the generalized Rosenau–Burgers equation. The first scheme is a two-level nonlinear Crank–Nicolson difference scheme and the second is a three-level linearized difference scheme. We derive the discrete mass and energy properties, and provide rigorous proofs for the boundedness, existence, and convergence with order (O(tau ^2 + h^s), (s = 4, 6, 8)) of these proposed generalized compact difference schemes, where (tau ) and h denote the time- and space-steps, respectively. Finally, the validity of the theoretical analysis is verified through numerical experiments, confirming the effectiveness of the proposed schemes.

The Muirhead Mean (MM) tools are more powerful and well-known operators utilized to express interrelationships among any input arguments considering different variables. Multi-attribute decision-making (MADM) technique is used to evaluate a reliable optimal option based on some realistic characteristics or criteria. The aggregation operators (AOs) play a crucial role in the aggregating and decision-making (DM) processes. In this article, we generalize the theory of intuitionistic fuzzy sets (IFSs) with Frank t-norm and t-conorm. Some robust operational laws of Frank t-norms and t-conorms are also expressed. By inspiring the significance and advantages of the MM operators, we derive some robust mathematical approaches, including intuitionistic fuzzy Frank Muirhead mean (IFFMM) and intuitionistic fuzzy Frank weighted Muirhead mean (IFFWMM) operators. By generalizing the concepts of Dual MM (DMM) operators, we establish a list of new methodologies such as intuitionistic fuzzy Frank Dual Muirhead mean (IFFDMM) and intuitionistic fuzzy Frank weighted Dual Muirhead mean (IFFWDMM). Some prominent characteristics and exceptional cases are discussed in detail. Furthermore, an algorithm is established to evaluate a MADM problem based on derived mathematical approaches. To examine the credibility and effectiveness of diagnosed approaches, we illustrate an experimental case study to assess a suitable optimal option from a group of options. To show the intensity and effectiveness of our derived approaches, a brief discussion about a comparative study is also presented, in which we compare the results of existing approaches with diagnosed mathematical approaches.

Partial order formal structure analysis (POFSA) is an emerging theory in the field of concept cognitive learning (CCL). Attribute partial order structure diagram (APOSD) is the visual expression of the knowledge structure in POFSA. It has the advantages of explicit expression of the hierarchies of attributes and concise visual expression of the knowledge structure. This paper mainly focuses on the incremental CCL of APOSD in dynamic data circumstances. Firstly, the concept of location information coding of nodes in APOSD is proposed to express the position of nodes in the entire diagram as well as the relationships between nodes, which is an important tool throughout this paper. Secondly, by analyzing the relationship between new objects and objects in the original diagram, dynamic learning strategy for APOSD is proposed. Thirdly, in order to balance the efficiency and accuracy of dynamic learning, a dynamic-static alternating self-learning method for APOSD is proposed, which is an improved incremental learning strategy. Finally, comparative experiments illustrate that compared with non-incremental learning method of APOSD and concept lattice, the two proposed incremental learning methods of APOSD can effectively achieve dynamic self-updating of the knowledge base when processing dynamic data, and provide another perspective for discovering knowledge from the same data. Besides, the effectiveness of the improved incremental learning strategy is verified as well.

We introduce the dual-Hilbert space and study the basic properties of a dual operator and its generalized inverse on this space. We provide upper bounds on the perturbation of the dual Moore–Penrose inverse of the dual operator if the dual operator is injective or surjective. If the null space or range space of the perturbed dual operator is invariant, stable perturbations are used to give the perturbation bounds for the dual Moore–Penrose inverse. Additionally, given the aforementioned conditions, perturbation bounds for the least squares solution are provided. The upper bounds on the distance between the solution of a perturbed least squares problem and the set of all of its unperturbed solutions under the dual operator norm are also presented.

A topological index is a numerical property of a molecular graph that explains structural features of molecules. The potential of topological indices to discriminate between distinct structures is a significant topic to investigate. In this context, the exponential degree-based indices were put forward in the literature. The present work focuses on the exponential augmented Zagreb index (EAZ), which is defined for a graph G as

where (d_i) represents the degree of the vertex (v_i)and E(G) denotes the edge set of G. This work characterizes the maximal unicyclic graph for EAZ in terms of graph order, which was posed as an open problem in the recent article Cruz et al. (MATCH Commun Math Comput Chem 88:481-503, 2022).

In this paper, various properties of core-EP matrices are investigated. We introduce the MPDMP matrix associated with A and by means of it, some properties and equivalent conditions of core-EP matrices can be obtained. Also, properties of MPD, DMP, and CMP inverses are studied and we prove that in the class of core-EP matrices, DMP, MPD, and Drazin inverses are the same. Moreover, DMP and MPD binary relation orders are introduced and the relationship between these orders and other binary relation orders are considered.

The goal of this investigation is to achieve the numerical solution of a two-dimensional parabolic partial differential equation(PDE). The proposed method of this paper is based on the discrete Legendre Galerkin method and spectral collocation method to simplify the spatial derivatives and time derivatives. The discrete Galerkin method is a very fast technique compared to the classical Galerkin method since a finite sum is needed for determining the interpolation coefficients. The operational matrix of the discrete Legendre polynomials is introduced to discretize the time derivatives. Using these couple of techniques and the collocation method, the aforementioned problem is transformed into a solvable algebraic system. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new method.