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The purpose of this paper is to propose a stable high-order absorbing boundary condition (ABC) based on new continued fraction for scalar wave propagation in 2D and 3D unbounded layers.

The ABC is obtained based on continued fraction (CF) expansion of the frequency-domain dynamic stiffness coefficient (DtN kernel) on the artificial boundary of a truncated infinite domain. The CF which has been used to the thin layer method in [69] will be applied to the DtN method to develop a time-domain high-order ABC for the transient scalar wave propagation in 2D. Furthermore, a new stable composite-CF is proposed in this study for 3D unbounded layers by nesting the above CF for 2D layer and another CF.

The ABS has been transformed from frequency to time domain by using the auxiliary variable technique. The high-order time-domain ABC can couple seamlessly with the finite element method. The instability of the ABC-FEM coupled system is discussed and cured.

This manuscript establishes a stable high-order time-domain ABC for the scalar wave equation in 2D and 3D unbounded layers, which is based on the new continued fraction. The high-order time-domain ABC can couple seamlessly with the finite element method. The instability of the coupled system is discussed and cured.

This paper aims to discuss the relationship between the continued fraction form of the analytical solution in the frequency domain, the orthogonal function expansion and their circuit realization to derive an efficient representation of the eddy-current field in the conducting sheet and wire/cylinder. Effective frequency ranges of representations are analytically derived.

The Cauer circuit representation is derived from the continued fraction form of analytical solution and from the orthogonal polynomial expansion. Simple circuit calculations give the upper frequency bounds where the truncated circuit and orthogonal expansion are applicable.

The Cauer circuit representation and the orthogonal polynomial expansions for the magnetic sheet in the E-mode and for the wire in the axial H-mode are derived. The upper frequency bound for the Cauer circuit is roughly proportional to N⁴ with N inductive elements, whereas the frequency bound for the finite element eddy-current analysis with uniform N elements is roughly proportional to N².

The Cauer circuit representation is expected to provide an efficient homogenization method because it requires only several elements to describe the eddy-current field over a wide frequency range.

The applicable frequency ranges are analytically derived depending on the conductor geometry and on the truncation types.

This paper aims to consider a single server queue with system disasters and impatience behavior are evident in our daily life. For this purpose, authors require to know the general behavior of these systems. Transient analysis shows for us how the system will operate up to some time instant t.

In this paper, authors consider a single server queue with system disaster and impatient behavior of customers in a multi-phase random environment, in which the system transits to a repair state after each system disaster. When the system is in a failure phase or going through a repair phase, the new arrivals would be impatient. In case the system is not repaired before the customer’s time expires, the customer would leave the queue and never return. Moreover, after repair, the system becomes ready for service in an operative phase with probability $q_{i} \ge 0.$. Using generating functions along with continued fractions and some properties of the confluent hypergeometric function, authors obtained on their own results.

Explicit expressions have been obtained for the time-dependent probabilities of the underlying queuing model. Also, time-dependent mean and variance of customers in the system are deduced.

The system authors are dealing with is somewhat complicated, there are some performance measures that cannot be achieved, but some of them have been obtained, such as the expectation and variance of the number of customers in the system.

Based on the obtained results, some numerical examples are some numerical examples are presented to illustrate the effect of various parameters on the behavior of the proposed system.

Authors’ studied transient analysis of a single server queue with system disaster and impatient customer system is suitable for behavior interpretation of many systems in our lives, such as telecommunication networks, inventory systems and impatient telephone switchboard customers, manufacturing system and service system.

To the best of the author’s/authors’ knowledge and according to the literature survey, in a multi-phase random environment, no previous published article is presented for transient analysis of a single server queue with system disaster and impatient customer behavior in a random environment.

Past work has shown that failure tolerance by principals has the potential to stimulate innovation, but has not examined how this affects which projects principals will start. We demonstrate that failure tolerance has an equilibrium price – in terms of an investor’s required share of equity – that increases in the level of radical innovation. Financiers with investment strategies that tolerate early failure will endogenously choose to fund less radical innovations, while the most radical innovations (for whom the price of failure tolerance is too high) can only be started by investors who are not failure tolerant. Since policies to stimulate innovation must often be set before specific investments in innovative projects are made, this creates a trade-off between a policy that encourages experimentation ex post and the one that funds experimental projects ex ante. In equilibrium, it is possible that all competing financiers choose to offer failure tolerant contracts to attract entrepreneurs, leaving no capital to fund the most radical, experimental projects in the economy. The impact of different innovation policies can help to explain who finances radical innovations, and when and where radical innovation occurs.

This chapter presents a brief description of the development of capitalism in Argentina, focusing on the situation of the working class and its practices. It analyzes the relationship between the main directions of capitalist development and the means of struggle used by the working class for more than a hundred years. It describes the predominant tendencies (in breadth and depth) of the development of capitalism in Argentina and the consequent main direction of the movement of the population (attraction or repulsion) in relation to capitalist relations. From the nineteenth century to the mid-seventies of the twentieth century, capitalism developed mainly in breadth, incorporating population, general strikes became a frequent practice and workers achieved a place in the institutional system. In the last quarter of the twentieth century, capitalist developed mainly in depth and, consequently, repulsion of population became dominant, increasing unemployment and poverty. Workers’ organizations lost some of their strength, but new organizations of the unemployed and the poor emerged, and roadblocks extended as an instrument of struggle.

The purpose of this paper is to detail the design of a fractional controller which was developed for the suppression of the flexural vibrations of the first mode of a smart beam.

During the design of the fractional controller, in addition to the classical control parameters such as the controller gain and the bandwidth; the order of the derivative effect was also included as another design parameter. The controller was then designed by considering the closed loop frequency responses of different fractional orders of Continued Fraction Expansion (CFE) method.

The first, second, third and fourth order approximations of CFE method were studied for the performance analysis of the controller. It was determined that the increase in the order resulted in better vibration level suppression at the resonance. The robustness analysis of the developed controllers was also conducted.

The experimentally obtained free and forced vibration results indicated that the increase in the order of the approximations yielded better performance around the first flexural resonance region of the smart beam and proved to yield better performance than the classical integer order controllers.

Evaluation of the performance of a developed fractional controller was realized by using different approach orders of the CFE method for the suppression of the flexural vibrations of a smart beam.

Computation have been made of the three‐dimensional flow field development, chemical reaction and combustion processes in a typical afterburner system under both isothermal and reacting flow conditions. The calculations are based upon a numerical solution of the time‐averaged transport equations for mass, momentum, turbulence kinetic energy, dissipation rate, enthalpy and species concentrations using a finite‐volume formulation. The physical models include the k—ε turbulence model, the eddy break‐up model, a two‐step reaction model, a droplet vaporization and combustion model and six‐flux radiation model. The mean flow structures are presented in important longitudinal and cross‐sectional planes which show certain striking similarities and contrasting differences for isothermal and reacting flows. The flame stabilizer flow is shown to be dominated by a complex combination of recirculation and vortex patterns. Combustion alters convergence and mixing flow patterns downstream of the flame stabilizer, thus influencing the selection of the fuel injection system. The predicted reacting flow parameters identify a number of design parameters such as fuel injector location, high degree reaction zone, nozzle opening area and the corresponding fuel flow rate.

The finite element method (FEM) is used to calculate the two-dimensional anti-plane dynamic response of structure embedded in D’Alembert viscoelastic multilayered soil on the rigid bedrock. This paper aims to research a time-domain absorbing boundary condition (ABC), which should be imposed on the truncation boundary of the finite domain to represent the dynamic interaction between the truncated infinite domain and the finite domain.

A high-order ABC for scalar wave propagation in the D’Alembert viscoelastic multilayered media is proposed. A new operator separation method and the mode reduction are adopted to construct the time-domain ABC.

The derivation of the ABC is accurate for the single layer but less accurate for the multilayer. To achieve high accuracy, therefore, the distance from the truncation boundary to the region of interest can be zero for the single layer but need to be about 0.5 times of the total layer height of the infinite domain for the multilayer. Both single-layered and multilayered numerical examples verify that the accuracy of the ABC is almost the same for both cases of only using the modal number excited by dynamic load and using the full modal number of infinite domain. Using the ABC with reduced modes can not only reduce the computation cost but also be more friendly to the stability. Numerical examples demonstrate the superior properties of the proposed ABC with stability, high accuracy and remarkable coupling with the FEM.

A high-order time-domain ABC for scalar wave propagation in the D’Alembert viscoelastic multilayered media is proposed. The proposed ABC is suitable for both linear elastic and D’Alembert viscoelastic media, and it can be coupled seamlessly with the FEM. A new operator separation method combining mode reduction is presented with better stability than the existing methods.

The purpose of this paper is to better understand the challenges involved in introducing Lesson Study (LS) into teacher education in Malawi by studying mathematics teacher educators’ (TEs’) understanding of planning for LS.

This study is a part of a wider ongoing project designed to improve the quality and capacity of mathematics teacher education in Malawi. One of its components is professional development of all mathematics TEs in Malawi using an LS model. The units being analyzed comprise of the TEs’ written lesson plans and qualitative content analysis is the chosen analytical approach.

Based on the analyzed research lesson plans, the TEs have difficulty in focusing on their own learning parallel to the student teachers’ (STs’) learning, and struggle with predicting STs’ responses to tasks. In addition, there is a pervasive lack of emphasis on planned and focused observation of STs’ learning, as evidenced by a review of the research lesson plans.

This is a small-scale study due to LS being introduced to Malawi teacher education for the first time and the need to test before possible upscaling.

The paper includes a description of mathematics TEs’ understanding of LS in an African context, which can be a valuable information for TEs who are attempting to use LS.

This paper fulfills an identified need to learn more about TEs’ understanding of LS worldwide.

Benchmark problem 13 of the TEAM Workshop consists of steel plates around a coil (a nonlinear magnetostatic problem). Seventeen computer codes developed by twelve groups are applied, and twenty‐five solutions are compared with each other and with experimental results. In addition to the numerical calculations, two theoretical presentations are given in order to explain discrepancies between the calculations and the experiment.