This 763 page book, available as both hardback and paperback, is unique in the sense that it sails across the ocean of contemporary advanced mathematics by an impressive tour de force. In twelve ambitious chapters it covers major topics in set theory, groups, rings and fields, vector spaces, matrix theory, linear functionals and special functions. The depth of coverage, however, is highly uneven. Two chapters on linear functionals, one chapter on inner products and norms, and one chapter on convergence in normed vector spaces contain a wealth of finely tuned knowledge. On the other hand, Chapter 1 on the foundations of computation, and the final Chapter 12 on special functions, just skim the surface. The selection of material and its presentation is evidently aimed at computer scientists and computer engineers. Their colleagues in other disciplines will find little that applies directly to their respective fields in spite of the many (more than 400) exercise problems. The author's approach is fundamentally different from similar titles in the literature, e.g. the time-honoured multi-edition text for physicists by Arfken. It provides a cogent, elegant and often esoteric structural framework for mathematical formulations, but with limited interest in exploring detailed solutions. A case in point is the brief presentation of Weber's Y (or N) function, better known as the Bessel function of the second kind. It appears as an eigenfunction of the Bessel operator (Section 12.4), but without any mention of how this function can be routinely eliminated from solutions where the physical quantity of interest is finite at the origin - a cornerstone of numerous practical problem solutions in physics and engineering. Other omissions are equally vexing. In Chapter 3 (Evaluation of Functions), the bisection method is dismissed in nine lines, and no mention is made of the secant method and Wegstein's method, which are just as important for numerical root finding as the analytical (and temperamental) Newton's method. Classical integration techniques are discussed, but not even one of the modern methods (e.g. Gaussian, Chebyshev, Rodau, Lobatto, Laguerre, Hermite and Filon integration) is given any mention. The classical Euler method for integrating ordinary differential equations is the subject of four pages, but there is nothing at all on the widely used Runge-Kutta techniques, nor the highly efficient Richardson's extrapolation method. Modified Bessel functions and Kelvin functions, essential tools in physics and engineering, are completely missing. Fourier transforms are bestowed a short section (11.8), but Laplace transforms? - nothing. Complex numbers pop up here and there, but complex calculus is ignored. The list continues.... Will this work become `... an ideal textbook for senior undergraduate and graduate students in the physical sciences and engineering...' and `... a valuable reference for working engineers...'? The author's claim for the affirmative is extremely tenuous, inasmuch as only highly motivated students with a broad command of mathematics would be able (if willing) to master its contents. The working engineer would most likely look for pragmatic texts, which provide practical/numerical examples in equally modern packing (e.g. Advanced Engineering Mathematics by Robert J Lopez (Addison Wesley, 2000)). Finally, a number of witty comments throughout the book lighten the task of its reading. Some readers will consider the statement (p 494) that `... eigenvalue is a mongrel word...' (in contrast, presumably, with the purebred Eigenwert and valeur propre) to be amusingly clever. An entire Section 8.2 (albeit consisting only of four lines) tells us that `... there seem to be almost as many ways to spell Chebyshev as there are written human languages...', then gracefully provides the proper Cyrillic spelling. Thomas Z Fahidy

Modern mathematical methods for physicists and engineers

This paper investigates the free vibrations of the rotating pretwisted functionally graded (FG) composite cylindrical panels reinforced with the graphene platelets (GPLs) by considering the cantilever boundary conditions. The weight fraction of the graphene platelets in each ply may be different, which leads to the layer-wise functionally graded composite cylindrical panels reinforced with the GPLs. The effective Young's modulus is calculated by the modified Halpin-Tsai model. The effective Poisson's ratio and mass density are derived by the rule of the mixture. The strain-displacement relationship is acquired by the Green strain tensor. Based on the first-order shear deformation theory, Chebyshev-Ritz method is used to obtain the natural frequencies of the rotating pretwisted functionally graded composite cylindrical panel reinforced with the GPLs. The natural frequencies are discussed by considering different material and geometry parameters of the rotating pretwisted functionally graded composite cylindrical panel reinforced with the GPLs, such as the GPL distribution pattern, the GPL weight fraction, the geometries of the GPLs, the pretwisted angle, the presetting angle and the rotating speed. Several validations are carried out, the numerical results are in good agreement with the results of the literature and ANSYS.

Free vibration of rotating pretwisted functionally graded composite cylindrical panel reinforced with graphene platelets

Nonlinear forced vibrations of functionally graded piezoelectric cylindrical shells under electric-thermo-mechanical loads

Vibration characteristics of rotating pretwisted composite tapered blade with graphene coating layers

This paper presents a general approach for free vibration analysis of spinning local elastic-foundation supported joined conical–cylindrical shells (JCCSs) with arbitrary boundary conditions. The Donnell’s shell theory is employed in the modeling of the structures, and all the effects of centrifugal forces, Coriolis forces as well as initial hoop tensions are considered. The approach utilizes the concept of artificial springs which permit convenient joining of the conical and cylindrical shells. Particularly, it can efficiently simulate arbitrary boundary conditions of JCCSs. The spinning JCCSs are locally or completely surrounded by elastic foundation, formulated by the Pasternak model. By using the orthogonal polynomial series as the admissible functions, the frequency equations of spinning JCCSs with arbitrary boundary conditions are derived using the Rayleigh–Ritz method. Then, some numerical examples are compared with the available literature and finite element results to validate the present approach. Finally, the effects of key parameters on free vibration characteristics are investigated including spring stiffness, spinning speed, cone angle and local elastic foundation.

A general approach for free vibration analysis of spinning joined conical–cylindrical shells with arbitrary boundary conditions

Free vibration of rotating cantilever pre-twisted panel with initial exponential function type geometric imperfection

A new structural dynamic model for the free vibration characteristic analysis of rotating pretwisted functionally graded (FG) sandwich blades is developed. The sandwich blade is made up of two functionally graded skins and a homogeneous material core. The thick shell theory is applied to derive the basic equations of motion of the rotating FG sandwich blade by considering the effects of centrifugal and Coriolis forces. The mode shapes are expanded in terms of two-dimensional algebraic polynomials in the Rayleigh–Ritz method, and the static and dynamic natural frequencies of the blade are obtained. The convergence analysis is studied, and the accuracy of the proposed model is verified by comparing with the literature results and ANSYS data. The effects of frequency parameters such as the twist angle, the thickness ratio, the aspect ratio, the layer thickness ratio, the scalar parameter of volume fraction, the stagger angle, and the rotation velocity on the vibration characteristics for pretwist FG sandwich blade are investigated in detail. In addition, the phenomena of frequency locus veering and mode shape exchanging occur in the static and dynamic states. Frequency locus veering is essentially caused by the coupling between different modes.

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Free Vibration Analysis of Rotating Pretwisted Functionally Graded Sandwich Blades

Free vibration and buckling of eccentric rotating FG-GPLRC cylindrical shell using first-order shear deformation theory

Buckling and free vibration of eccentric rotating CFRP cylindrical shell base on FSDT

The modeling and forecasting of BBS (Bulletin Board System) posts time series is crucial for government agencies, corporations and website operators to monitor public opinion. Accurate prediction of the number of BBS posts will assist government agencies or corporations in making timely decisions and estimating the future number of BBS posts will help website operators to allocate resources to deal with the possible hot events pressure. By combining sample entropy (SampEn) and deep neural networks (DNN), an approach (SampEn-DNN) is proposed for BBS posts time series modeling and forecasting. The main idea of SampEn-DNN is to utilize SampEn to decide the input vectors of DNN with smallest complexity, and DNN to enhance the prediction performance of time series. Selecting Tianya Zatan new posts as the data source, the performances of SampEn-DNN were compared with auto-regressive integrated moving average (ARIMA), seasonal ARIMA, polynomial regression, neural networks, etc. approaches for prediction of the daily number of new posts. From the experimental results, it can be found that the proposed approach SampEn-DNN outperforms the state-of-the-art approaches for BBS posts time series modeling and forecasting.

https://www.mdpi.com/1099-4300/21/1/57/pdf

L. T. Liu

Papers

Free vibration of rotating cantilever pre-twisted panel with initial exponential function type geometric imperfection

Free Vibration Analysis of Rotating Pretwisted Functionally Graded Sandwich Blades

Free vibration and buckling of eccentric rotating FG-GPLRC cylindrical shell using first-order shear deformation theory

Buckling and free vibration of eccentric rotating CFRP cylindrical shell base on FSDT

BBS Posts Time Series Analysis based on Sample Entropy and Deep Neural Networks.