Experiences in Computing Mesh Root Systems for Dynkin Diagrams Using Maple and C++

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.

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.

Coxeter spectral classification of almost TP-critical one-peak posets using symbolic and numeric computations

Analytical SN solutions in heterogeneous slabs using symbolic algebra computer programs

Using noncommutative Gröbner bases in solving partially prescribed matrix inverse completion problems

A Review of Arithmetic Circuit Verification Using Symbolic Computer Algebra

We derive the full set of beta functions for the marginal essential couplings of projectable Ho\ifmmode \check{r}\else \v{r}\fi{}ava gravity in ($3+1$)-dimensional spacetime. To this end we compute the divergent part of the one-loop effective action in static background with an arbitrary spatial metric. The computation is done in several steps: reduction of the problem to three dimensions, extraction of an operator square root from the spatial part of the fluctuation operator, and evaluation of its trace using the method of universal functional traces. This provides us with the renormalization of couplings in the potential part of the action which we combine with the results for the kinetic part obtained previously. The calculation uses symbolic computer algebra and is performed in four different gauges yielding identical results for the essential beta functions. We additionally check the calculation by evaluating the effective action on a special background with spherical spatial slices using an alternative method of spectral summation. We conclude with a preliminary discussion of the properties of the beta functions and the resulting renormalization group flow, identifying several candidate asymptotically free fixed points.

Multiplier circuit covers a more extensive area of embedded system application in digital signal processing, cryptography, and multimedia. Nonstandard implementations and custom optimization are being done to reduce the size of multipliers. The circuit became prone to a buggy, and hence the demand for verification increased. Formal verification methods, such as satisfiability (SAT), symbolic computer algebra (SCA), and binary decision diagrams (BDDs) have made massive progress over the last few decades. However, these methods are insufficient to verify the optimized multipliers. SAT-based equivalence checking is computationally expensive. SCA-based backward rewriting is limited to algebraic-friendly multipliers. The complexity of BDDs is exponential with the input size. Although, by allowing an additional variable method, the size of the BDD is limited to 4th degree polynomial of the number of the inputs, this method is not explored to verify optimized multipliers. This article focus on verifying integer multipliers with diverse architectures. We propose an algorithm for the direct construction of BDDs without traversing circuits and generate BDDs up to 1024 bits. We utilize the additional variable method and constructing BDDs using high-to-low variable ordering. We reduce the complexity of BDD size to a 3rd degree polynomial. We generate BDDs and verify the multipliers with various architectures up to 64 bits. We propose a method to verify optimized multipliers by checking equivalence and verifying up to 32-bits optimized multipliers. We do the error tolerance analysis of our approach by inserting bugs in a circuit at various locations.

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 aim of SNC 2009 is to offer a forum for researchers in symbolic and numeric computation to present their work, interact, exchange ideas and identify important problems in this research area. SNC 2009 continues a successful tradition of previous highly successful workshops in the area of symbolic and numeric computation: SNAP 96, held July 15-17, 1996 in Sophia Antipolis, FranceSNC 2005, held July 19-21, 2005 in Xi'an, ChinaSNC 2007, held July 25-27, 2007 in London Ontario, Canada A warm thank you goes to all those that worked hard to make SNC 2009 happen and in particular the local organizers, the program committee members, the anonymous referees, the invited speakers and the participants. The SNC 2009 Call For Papers solicited submissions in several topics: Hybrid symbolic-numeric algorithmsApproximate polynomial GCD and factorizationSymbolic-numeric methods for solving polynomial systemsResultants and structured matrices for symbolic-numeric computationDifferential equations for symbolic-numeric computationSymbolic-numeric methods for geometric computationSymbolic-numeric algorithms in algebraic geometrySymbolic-numeric algorithms for nonlinear optimizationNumeric computation of characteristic sets and Groebner basesImplementation of symbolic-numeric algorithmsApproximate algebraic algorithmsApplications of symbolic-numeric computation SNC 2009 is sponsored by the University of Tsukuba (http://www.tsukuba.ac.jp/english/). SNC 2009 is in cooperation with ACM SIGSAM (http://www.sigsam.org/) and we wish to thank the Chair of SIGSAM, Dr. Mark Giesbrecht, for his continuous help and support. SNC 2009 is also in cooperation with JSSAC, the Japan Society for Symbolic and Algebraic Computation (http://www.jssac.org/). We hope that the current book of proceedings of SNC 2009 will become a useful resource for researchers in Symbolic-Numeric Computation and other related research areas. We also hope that it will become another testament of the liveliness and vibrancy of this research area and a precursor of the future developments that await us ahead.

In recent years, formal methods based on Symbolic Computer Algebra (SCA) have shown very good results in verification of integer multipliers. The success is based on removing redundant terms (vanishing monomials) early which allows to avoid the explosion in the number of monomials during backward rewriting. However, the SCA approaches still suffer from two major problems: (1) high dependence on the detection of Half Adders (HAs) realized as AND-XOR gates in the multiplier netlist, and (2) extremely large search space for finding the source of the vanishing monomials. As a consequence, if the multiplier consists of dirty logic, i.e. for instance using non-standard libraries or logic optimization, the existing SCA methods are completely blind on the resulting polynomials, and their techniques for effective division fail. In this paper, we present RevSCA. RevSCA brings back light into backward rewriting by identifying the atomic blocks of the arithmetic circuits using dedicated reverse engineering techniques. Our approach takes advantage of these atomic blocks to detect all sources of vanishing monomials independent of the design architecture. Furthermore, it cuts the local vanishing removal time drastically due to limiting the search space to a small part of the design only. Experimental results confirm the efficiency of our approach in verification of a wide variety of integer multipliers with up to 1024 output bits.

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.

Mathematical models of biochemical reaction networks involve nonlinear differential equations with a number of parameters whose values are generally not known a priori. Analysis of such networks includes both analytical and numerical work. The analytical part of such analysis often requires manipulation with massive algebraic expressions that may become challenging due to the inherent nonlinearities and unknown parameters in the model. Computer algebra systems (CASs), which are also capable of performing calculations with symbols and to generate ready‐to‐use computer codes for numerical computations, are excellent tools in such analysis. The present work gives a brief description of how we can benefit from computer algebra to overcome such difficulties and combine symbolic and numeric computation techniques in such analysis with several illustrative biochemical reaction networks. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

On an application of symbolic computation and computer graphics to root-finders: The case of multiple roots of unknown multiplicity

What task should be played by symbolic mathematical computation facilities in Pure and applied Mathematics, scientific and engineering “problem solving environments”? Drawing upon standard facilities such as numerical and graphical libraries, symbolic computation should be useful for the creation and manipulation of mathematical models. Symbolic representation and manipulation can potentially play a central organizing role in PSEs since their more general object representation allows a program to deal with a wider range of computational issues. Symbolic processing power of software as MathCAD (MathCAD explorer version 8) is used to demonstrate the basics as well as its application dispersion of thermoelastic waves.