Analytical SN solutions in heterogeneous slabs using symbolic algebra computer programs

The International Symposium on Symbolic and Algebraic Computation (ISSAC) is the premier conference for research in symbolic computation and computer algebra. ISSAC 2012 is the 37th meeting in the series, started in 1966 and held annually since 1981, in North America, Europe and Asia. The conference presents a range of invited speakers, tutorials, poster sessions, software demonstrations and vendor exhibits with a centerpiece of contributed research papers.

Non-equilibrium molecular dynamics integrators using Maple

Modern functional programming languages and lazy functional techniques are useful for describing and implementing abstract mathematical objects in quantum mechanics. Scientists can use them both for pedagogical purposes and for real, not too computationally intensive, but conceptually and algorithmically difficult applications. This article shows how to perform simple abstract computations on state vectors and discusses the construction of lazy algorithms that enormously simplify manipulation of potentially infinite data structures or iterative processes. Lazy functional techniques can often replace the use of symbolic computer algebra packages, while also offering an interesting algorithmic complement to the manipulation of mathematical data. These techniques are more efficient than blindly used symbolic algebra and are easy to integrate with the numerical code.

The authors propose a method for the determination of singularities in motion and displacement functions for a seven degree‐of‐freedom manipulator. The manipulator is considered, hereby, as a set of six degree‐of‐freedom manipulators. It is proven that two types of singularities in motion can occur at link positions that are independent and dependent with respect to the trajectory to be executed. The relations between the structure of singularity equations and displacement equations are discussed. The derivation of displacement equations for a manipulator with singularities of the second type is based on the idea of modeling of a manipulator by two open kinematic chains and invariants that may be found for these chains. The inverse kinematic equations and the equations of singularities in motion have been derived using a symbolic computer program which can handle manipulators of general structure (with five, six, and seven degrees‐of‐freedom). This program is written in the Symbolic Computation Language REDUCE.

Maple is a computer algebra system and programming language that can be used for numerical computations, solving equations, manipulating symbolic expressions, plotting, and programming, just to name a few of the basics. APPL is, simply, a set of supplementary Maple commands and procedures that augments the existing computer algebra system. In effect, APPL takes the capabilities of Maple and turns it into a computer algebra system for computational probability. This chapter contains guidelines for using Maple, and discusses the Maple commands that are used in APPL programming. After reading this chapter, an APPL user will have the knowledge necessary to modify the APPL code to meet his or her particular needs. We will start with a discussion of basic numeric computation, then advance to defining variables, symbolic computations, functions, data types, solving equations, calculus and graphing. Then we will discuss the programming features of Maple that facilitate building the APPL language: loops, conditions, and procedures.

We show that the reduced mesh root systems and mesh geometries of roots for each of the simply-laced Dynkin diagrams can be classified by applying symbolic computer algebra computations and numeric algorithmic computations in Maple, C++ and Linux. Results of our computing experiences are presented in a table of Section 2 and in Section 6.

We continue and complete a Coxeter spectral study (presented in our talk given in SYNASC11, Timisoara, September 2011 [6]) of the root systems in the sense of Bourbaki [4], the mesh geometries Γ(R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ</sub> , Φ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</sub> ) of roots of Δ in the sense of [20], and matrix morsifications A ∈ Mor <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ</sub> , for simply-laced Dynkin diagrams Δ ∈ {A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , D <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6</sub> , E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">7</sub> , E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</sub> }. One of the main aims of the talk is to present a progress obtained during the last year in solving the problems stated in our talk [6]. We show that the reduced mesh root systems and mesh geometries of roots for each of the Dynkin diagrams Δ can be classified for n ≤ 9 by applying symbolic computer algebra computations and numeric algorithmic computations. We also compute the isotropy group Gl(n, Z) <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ</sub> of Δ defined in [22]-[23], determine its structure and compute the Gl(n, Z) <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ</sub> -orbits of the morsifications A ∈ Mor <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ</sub> , for Δ ∈ {A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , D <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> }, with n ≤ 9 and 4 ≤ m ≤ 8. Results of our computing experiences are presented in three tables of Section 4.

Digital technologies are changing the teaching and learning of algebra. This chapter introduces the MEx platform, a digital environment specifically designed to support symbolic algebra structure sense. The first part of the chapter reviews different approaches to using digital tools for teaching and learning algebra. This review includes interface design, game-based learning, multi-representational learning, and symbolic algebra support. The second part describes a range of digital tools that can be employed to develop structure sense, including Desmos, Graspable Math, Mathigon, Khan Academy, and Alcumus. The third part of the chapter outlines design principles for supporting structure sense: adaptivity, gamification, interaction design, learning analytics dashboards, and worked examples. The fourth part introduces the MEx platform, built upon algebraic ideas essential for developing structure sense, including equivalence, substitution, and the dual nature of algebraic expressions as processes and objects. The design principles of MEx include metaphors, levelling-up, symbolic computation, feedback, visual salience, variation, and gamification.

Automatic modeling of mechatronic systems via symbolic approach

To help along the analysis and design of nonlinear control systems the NONLINCON package, an acronym for nonlinear control, was developed. This note addresses the usefulness of symbolic computation, and of the NONLINCON package in particular, for the symbolic analysis and design of nonlinear control systems. The symbolic computation program MAPLE is used as computing substratum. Textbook problems show that the NONLINCON package can be used successfully. A larger scale problem is too complex, however, to be solved with the current versions of NONLINCON and MAPLE. The conclusion is that symbolic computation is a viable approach for textbook problems, but not yet for more complex ones. >

This chapter describes the role of symbolic computation in several phases of the control system design cycle, namely modelling, analysis and synthesis. It provides some mathematical background of nonlinear system theory and presents a guided tour through those phases using typical examples, both contrived and real life. To name them, the symbolic computation of (i) the zero dynamics, used for different purposes, (ii) the input-output exact linearising state feedback, and (iii) the state-space exact linearising state feedback, all for nonlinear dynamic systems, are discussed. The examples come from a diverse range of applications and provide a picturesque landscape of this area of research. The possibilities and limitations of the so-called NON^CON package, based on the symbolic computation program Maple, are outlined with these examples. This package was developed to aid with analytical computations in the area of nonlinear control systems.

Streamline models are routinely used for waterflood optimization and management, and are being extended to more complex processes, e.g., compositional simulation. Despite these new developments, no systematic study has examined the underlying numerical spatial and temporal discretization errors in streamline simulation and their convergence. Such studies are a prerequisite to determining the optimal density of streamlines during simulation and ensuring the resulting accuracy of the solution. In this paper we first examine transverse spatial errors, e.g., errors due to the number or placement of streamlines. We provide an analytic proof and a numeric demonstration of the order of spatial convergence of the mass balance discretization error. Both global and local calculations are performed, and demonstrate the impact of stagnation regions on the order of convergence. A second transverse error arises for faulted grids, where lack of flux continuity at cell faces can lead to incorrect trajectories. These trajectory errors are of zeroth order, and can only be resolved by introducing additional degrees of freedom into the streamline velocity model. Longitudinal spatial errors also arise, and are associated with the inaccurate calculation of time of flight across cells. We show that the commonly used algorithm for corner point cells leads to inaccurate time of flight calculations for stratigraphic grids, depending upon aspect ratio. We provide a simple and exact means of calculating the time of flight for arbitrary corner point cells, or unstructured grids, in two or three dimensions, for either compressible or incompressible flow. Finally, using this new time of flight formulation, we analyze a series of cross-sectional finite difference simulations to identify grid orientation errors in the numerical calculation of flux and spatial error.

On the application of the analytical discrete ordinates method to the solution of nonclassical transport problems in slab geometry

Analytic derivation of the correction factor for the improved coarse mesh method

It is known, that i.r. of double point groups can be expressed in terms of Wigner functions. Using the projection method proposed by O. Barut and R. Raczka the appropriate basis of i.r. of double point groups within the harmonic oscillator space have been derived. The appropriate numeric and symbolic algebra programs have been constructed.