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The Bounded and Precise Word Problems for Presentations of Groups
We introduce and study the bounded word problem and the precise word problem for groups given by means of generators and defining
relations. For example, for every finitely presented group, the bounded word problem is in
eBook
Numerical analysis of liquid-solid adsorption model
In this paper, the numerical algorithms for solution of pore volume and surface diffusion model of adsorption systems are constructed and investigated. The approximation of PDEs is done by using the finite volume method for space derivatives and ODE15s solvers for numerical integration in time. The analysis of adaptive in time integration algorithms is presented. The main aim of this work is to analyze the sensitivity of the solution with respect to the main parameters of the mathematical model. Such a control analysis is done for a linearized and normalized mathematical model. The obtained results are compared with simulations done for a full nonlinear mathematical model.
Journal Article
Randomization, relaxation, and complexity in polynomial equation solving : Banff International Research Station workshop on Randomization, Relaxation, and Complexity, February 28-March 5, 2010, Banff, Ontario, Canada
This volume corresponds to the Banff International Research Station Workshop on Randomization, Relaxation, and Complexity, held from February 28-March 5, 2010 in Banff, Ontario, Canada. This volume contains a sample of advanced algorithmic techniques underpinning the solution of systems of polynomial equations. The papers are written by leading experts in algorithmic algebraic geometry and touch upon core topics such as homotopy methods for approximating complex solutions, robust floating point methods for clusters of roots, and speed-ups for counting real solutions. Vital related topics such as circuit complexity, random polynomials over local fields, tropical geometry, and the theory of fewnomials, amoebae, and coamoebae are treated as well. Recent advances on Smale's 17th Problem, which deals with numerical algorithms that approximate a single complex solution in average-case polynomial time, are also surveyed.
eBook
An analytic approach for optimal coordination of overcurrent relays
This study presents a new analytic approach to optimal coordination of directional overcurrent relays. This approach is based on the selection of optimum pickup current and time dial setting, in order to obtain minimum operating time for the relays, while satisfying various coordination and boundary constraints. Based on the new optimal relay setting procedure, an iterative numerical solution is proposed. The proposed numerical algorithm converges to the global optimum values, which are independent of initial values and the order of relay setting. The proposed method is applied to three different test systems. The new method is compared with some previously proposed analytic and evolutionary approaches. The results demonstrate the advantages of the proposed method over the previous works.
Journal Article
Exascale applications: skin in the game
As noted in Wikipedia, skin in the game refers to having ‘incurred risk by being involved in achieving a goal’, where ‘ skin is a synecdoche for the person involved, and game is the metaphor for actions on the field of play under discussion’. For exascale applications under development in the US Department of Energy Exascale Computing Project, nothing could be more apt, with the skin being exascale applications and the game being delivering comprehensive science-based computational applications that effectively exploit exascale high-performance computing technologies to provide breakthrough modelling and simulation and data science solutions. These solutions will yield high-confidence insights and answers to the most critical problems and challenges for the USA in scientific discovery, national security, energy assurance, economic competitiveness and advanced healthcare. This article is part of a discussion meeting issue ‘Numerical algorithms for high-performance computational science’.
Journal Article
A general numerical algorithm for nonlinear differential equations by the variational iteration method
Purpose
The purpose of this paper is to suggest a general numerical algorithm for nonlinear problems by the variational iteration method (VIM).
Design/methodology/approach
Firstly, the Laplace transform technique is used to reconstruct the variational iteration algorithm-II. Secondly, its convergence is strictly proved. Thirdly, the numerical steps for the algorithm is given. Finally, some examples are given to show the solution process and the effectiveness of the method.
Findings
No variational theory is needed to construct the numerical algorithm, and the incorporation of the Laplace method into the VIM makes the solution process much simpler.
Originality/value
A universal iteration formulation is suggested for nonlinear problems. The VIM cleans up the numerical road to differential equations.
Journal Article
Nonlinear optical properties and applications of 2D materials: theoretical and experimental aspects
In this review, we survey the recent advances in nonlinear optics and the applications of two-dimensional (2D) materials. We briefly cover the key developments pertaining to research in the nonlinear optics of graphene, the quintessential 2D material. Subsequently, we discuss the linear and nonlinear optical properties of several other 2D layered materials, including transition metal chalcogenides, black phosphorus, hexagonal boron nitride, perovskites, and topological insulators, as well as the recent progress in hybrid nanostructures containing 2D materials, such as composites with dyes, plasmonic particles, 2D crystals, and silicon integrated structures. Finally, we highlight a few representative current applications of 2D materials to photonic and optoelectronic devices.
Journal Article
Numerical algorithms for high-performance computational science
A number of features of today’s high-performance computers make it challenging to exploit these machines fully for computational science. These include increasing core counts but stagnant clock frequencies; the high cost of data movement; use of accelerators (GPUs, FPGAs, coprocessors), making architectures increasingly heterogeneous; and multi- ple precisions of floating-point arithmetic, including half-precision. Moreover, as well as maximizing speed and accuracy, minimizing energy consumption is an important criterion. New generations of algorithms are needed to tackle these challenges. We discuss some approaches that we can take to develop numerical algorithms for high-performance computational science, with a view to exploiting the next generation of supercomputers. This article is part of a discussion meeting issue ‘Numerical algorithms for high-performance computational science’.
Journal Article
Geometrical shape optimization in fluid mechanics using FreeFem
In this article, we present simple and robust numerical methods for two-dimensional geometrical shape optimization problems, in the context of viscous flows driven by the stationary Navier-Stokes equations at low Reynolds number. The salient features of our algorithm are exposed with an educational purpose; in particular, the numerical resolution of the nonlinear stationary Navier-Stokes system, the Hadamard boundary variation method for calculating the sensitivity of the minimized function of the domain, and the mesh update strategy are carefully described. Several pedagogical examples are discussed. The corresponding program is written in the FreeFem++ environment, and it is freely available. Its chief features—and notably the implementation details of the main steps of our algorithm—are carefully presented, so that it can easily be handled and elaborated upon to deal with different, or more complex physical situations.
Journal Article