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In this letter, based on the infinitesimal transformations with respect to the generalized coordinates and generalized momentums, we obtain the definition, determining equations and structure equation of the momentum‐dependent symmetry for the systems. This study directly leads to the non‐ Noether type conserved quantity for the systems. Further we also give the inverse issue of the momentum‐dependent symmetries of the systems. However, a theory of momentum‐dependent symmetries of the nonconservative Hamiltonian systems is established. Finally, an example is discussed to illustrate the results.

The purpose of this paper is to find new non-traveling wave solutions and study its localized structure of Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation.

The authors apply the Lie group method twice and combine with the Exp-function method and Riccati equation mapping method to the (2+1)-dimensional CDGKS equation.

The authors have obtained some new non-traveling wave solutions with two arbitrary functions of time variable.

As non-linear evolution equations is characterized by rich dynamical behavior, the authors just found some of them and others still to be found.

These results may help the authors to investigate some new localized structure and the interaction of waves in high-dimensional models. The new non-traveling wave solutions with two arbitrary functions of time variable are obtained for CDGKS equation using Lie group approach twice and combining with the Exp-function method and Riccati equation mapping method by the aid of Maple.

The purpose of this paper is to demonstrate new properties of continuous‐ and discrete‐time dynamical systems.

First, definitions of two types of spatial symmetry are introduced. These are used as definitions, which, along with existing knowledge show that it is possible to identify properties of dynamical systems that were previously unknown.

The main result of the paper is a new theorem regarding new properties of continuous‐ and discrete‐time dynamical systems.

The present study provides a starting point for further research on the differences between continuous‐ and discrete‐time dynamical systems. This work builds on the definition of spatial symmetry.

The theorem proved in this paper and the new properties of dynamical systems can be used to introduce new methods of approximating continuous‐time dynamical systems by discrete‐time dynamical systems and vice versa. Such approaches can also be helpful in constructing chaotic sources to model noise.

This paper offers contributions to the broader discussion of differences between continuous‐ and discrete‐time dynamical systems. In particular, the paper supports the statement that many discrete‐time processes cannot be embedded into continuous ones.

The purpose of this paper is to discuss the homoclinic breathe-wave solutions and the singular periodic solutions for (2 + 1)-dimensional generalized shallow water wave (GSWW) equation.

The Hirota bilinear method, the Lie symmetry method and the non-Lie symmetry method are applied to the (2 + 1)D GSWW equation.

A reduced (1 + 1)D potential KdV equation can be derived, and its soliton solutions are also presented.

As a typical nonlinear evolution equation, some dynamical behaviors are also discussed.

These results are very useful for investigating some localized geometry structures of dynamical behaviors and enriching dynamical features of solutions for the higher dimensional systems.

The framework, panorama, related applications and development of pansystems thinking are introduced and some problems concerning generalized systems, derivatives, symmetry and quantification are investigated, a sort of difference‐identity theory within the framework of pansystems thinking is cultivated, and a new type of clustering method with many intertransformations among some important pansystems concepts is presented. Related contents include: pansystems combination of mathematics, systems, dialectics and poetry thinking, methodology to comprehend laws by pansymmetries or by moving networks, the pansystems views of generalized order, quotientization and series‐parallel relations, control‐reachability, duality transformation, discoupling principle, communication theorems, general parental relations, etc.

It is very important to create a useful cyclic symmetric model for the investigation of the vibration reduction effect of hard-coating blisk. This study aims to develop a cyclic symmetry algorithm which can determine the mode of blisk in the sector coordinate system directly.

Using the exponential and real quasi-equivalent Fourier matrices, the formulas for solving the sector mode were derived, and the relationship between the two kinds of sector modes was also discussed. Based on the proposed cyclic symmetry algorithm, the vibration characteristics of an academic blisk were solved, and the formulas for solving the natural characteristics and vibration responses of the coated blisk were given.

A blisk with NiCrAlCoY+YSZ hard coating on both sides of each blade was chosen as a case to demonstrate the presented method. Based on the verification analysis model, the influences of coating thickness on the vibration reduction effect of the blisk were discussed. The results show that the hard coating has good vibration reduction effect on the blisk even the coating thickness is very thin and the vibration reduction effect of hard coating in the high frequency range is obviously better than that in the low frequency range.

As a large number of reduced order modeling methods of blisk are implemented based on the sector modes, the proposed method which can obtain the sector modes directly will significantly improve the efficiency of dynamic modeling and analysis of the coated blisk structure.

The purpose of this paper is to improve the GM(1, 1) model based on concave sequences.

First, the restored sequence of the GM(1, 1) model is proved to be convex, and the residual characters of the GM(1, 1) model for concave sequences are analyzed. Second, two symmetry transformations are introduced to transform an original concave sequence into a convex sequence, and then the GM(1, 1) model is established based on the convex sequence.

Compared with the traditional modeling method, the new method has high accuracy and is applicable for all concave sequence modeling.

Two cases are used to illustrate the superiority of this modeling method. Case A is to predict China’s per capita natural gas consumption, and case B is to predict the annual output of an oilfield.

The application scope of GM (1, 1) model is greatly extended.

This paper aims to deal with two-dimensional magneto-hydrodynamic (MHD) Falkner–Skan boundary layer flow of an incompressible viscous electrically conducting fluid over a permeable wall in the presence of a magnetic field.

Using the Lie group approach, the Lie algebra of infinitesimal generators of equivalence transformations is constructed for the equation under consideration. Using these suitable similarity transformations, the governing partial differential equations are reduced to linear and nonlinear ordinary differential equations (ODEs). Further, Haar wavelet approach is applied to the reduced ODE under the subalgebra 4.1 for constructing numerical solutions of the flow problem.

A new type of solutions was obtained of the MHD Falkner–Skan boundary layer flow problem using the Haar wavelet quasilinearization approach via Lie symmetric analysis.

To find a solution for the MHD Falkner–Skan boundary layer flow problem using the Haar wavelet quasilinearization approach via Lie symmetric analysis is a new approach for fluid problems.

A computational procedure is presented for the re‐analysis of large unsymmetric structural systems. The procedure is based on a novel partitioning strategy in which the responses of both the original and modified structures are approximated by linear combinations of symmetric and antisymmetric response vectors (or modes), each obtained by using a fraction of the degree of freedom of the finite element model of the structure. The other key elements of the procedure are: (a) lumping of the large number of design variables into a single tracing parameter; (b) operator splitting or restructuring of the governing finite element equations to delineate the symmetric and antisymmetric vectors constituting the responses of the original and modified structures; and (c) a stable and efficient iterative process for generating the response of the modified structure. The re‐analysis procedure is applied to linear static analysis of framed structures. Design modifications consisted of removing members resulting in topologically unsymmetric structures. The potential of the procedure on multiprocessor computers is discussed and its effectiveness is demonstrated by means of two numerical examples.

This paper presents a new simplified text of some concepts of pansystems methodology and related applications to pedagogy, methods of teaching, study and creation, including certain principles of operations research, systems theory, cybernetics, etc.